♥ T❤❡ ♣r♦♣❡rt✐❡s Kh

♥ T❡r✲❡①❛♠♣•❡s ✐♥ L

✼✼ ✸✳✺✳✸

R✲❜♦r♥✐t✉❞❡ ❞❡ ❢❛♠✐••❡s ❞❡ ❢♦♥❝t✐♦♥s✱ ✉♥ ♣r♦❜•è♠❡ q✉✐ ❡st ❡ss❡♥t✐❡• •♦rsq✉❡ •✬♦♥ ❡ss❛✐❡ ❞❡ ❝♦♠♣r❡♥❞r❡ •❡s ♣r♦♣r✐étés ❞✉ ❝❛•❝✉• ❢♦♥❝t✐♦♥♥❡• s✉r •❡s ❡s♣❛❝❡s L p ✳ ◆♦t♦♥s q✉❡ •❡ t❡r♠❡ ❞❡ ❣❛✉❝❤❡ ❞❡ •✬éq✉✐✈❛•❡♥❝❡ ♣❡✉t êtr❡ ❞é✜♥✐ ❞❛♥s t♦✉t ❡s♣❛❝❡ ❞❡ ❇❛♥❛❝❤ X ♣•✉tôt q✉❡ L p (Ω) ❡♥ ✈♦②❛♥t i≥0 f i ξ i ❝♦♠♠❡ ✉♥ é•é♠❡♥t ❞❡ L p ((0, 1), X)✱ ❡t ✉♥❡ ✈❡rs✐♦♥ ❞❡s ✐♥é❣❛•✐tés ❞❡ ❑❤✐♥t❝❤✐♥❡ ✭✐♥é❣❛•✐tés ❞❡ ❑❤✐♥t❝❤✐♥❡✲❑❛❤❛♥❡✮ s✬❛♣♣•✐q✉❡♥t à ❝❡ ♥✐✈❡❛✉ ❞❡ ❣é♥ér❛•✐té✳ P•✉s ♣ré❝✐sé♠❡♥t✱ ♣♦✉r t♦✉t❡ s✉✐t❡ ✜♥✐❡ (x i ) i≥0 ❞❛♥s X✱ ❡t t♦✉t p ∈ (1, ∞)✱ i≥0

x i ξ i Lp((0,1),X) ≈ cp i≥0 f i ξ i L2((0,1),X)

❈❡♣❡♥❞❛♥t✱ ♥♦✉s s♦♠♠❡s ❞✬❛✈❛♥t❛❣❡ ✐♥tér❡ssés✱ ❞❛♥s •❡ ❝♦♥t❡①t❡ ❞❡ •✬✐♥té❣r❛t✐♦♥ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡✱ à tr♦✉✈❡r ✉♥ ❛♥❛•♦❣✉❡ ❞✉ t❡r♠❡ ❞❡ ❞r♦✐t❡ ❞❡ ✭✶✮✱ q✉✐ ❡st ♣•✉s ❢❛❝✐•❡ à ❝❛•❝✉•❡r ♦✉ ♠❛•♦r❡r q✉❡ •❡ t❡r♠❡ ❞❡ ❣❛✉❝❤❡✳ ◆♦t♦♥s t♦✉t ❞❡ s✉✐t❡ q✉❡ •❛ ♠ê♠❡ ❢♦r♠✉•❡ ♥❡ ♣❡✉t ♣❛s ❢♦♥❝t✐♦♥♥❡r ❞❛♥s •❡ ❝❛❞r❡ ♥♦♥✲❝♦♠♠✉t❛t✐❢ ❝❛r ✐• ② ❛ ❞❡✉① ♥♦t✐♦♥s ♣♦ss✐❜•❡s ❞❡ ❢♦♥❝t✐♦♥s ❝❛rrés ✭•✐❣♥❡ ❡t ❝♦•♦♥♥❡✮ q✉✐ ♥❡ s♦♥t ♣❛s éq✉✐✈❛•❡♥t❡s ❞❛♥s L p ❞ès q✉❡ p = 2✳ ❙♦✐t M ✉♥ ❡s♣❛❝❡ ♠❡s✉ré ♥♦♥✲ ❝♦♠♠✉t❛t✐❢✱ M c ⊂ M •❡s ♦♣ér❛t❡✉rs à s✉♣♣♦rt ✜♥✐ ❡t S(M) •✬❡♥s❡♠❜•❡ ❞❡s s✉✐t❡s ✜♥✐❡s ❞❡ M c ✳ ❖♥ r❛♣♣❡••❡ q✉✬✉♥ ❡s♣❛❝❡ s②♠étr✐q✉❡ ❡st ✉♥ ❡s♣❛❝❡ q✉❛s✐✲ ❇❛♥❛❝❤ ❞❡ ❢♦♥❝t✐♦♥s t❡• q✉❡ •❛ ✭q✉❛s✐✲✮♥♦r♠❡ ♥❡ ❞é♣❡♥❞❡ q✉❡ ❞❡ •❛ ❞✐str✐❜✉t✐♦♥ ❞✉ ♠♦❞✉•❡ ❞❡ •❛ ❢♦♥❝t✐♦♥✳ P✉✐sq✉❡ •✬♦♥ ♣❡✉t ❞é✜♥✐r •❛ ❞✐str✐❜✉t✐♦♥ ❞✉ ♠♦❞✉•❡ ❞✬✉♥ ♦♣ér❛t❡✉r✱ ❝❡s ❡s♣❛❝❡s ❛❞♠❡tt❡♥t ❞❡s ❣é♥ér❛•✐s❛t✐♦♥s ♥♦♥✲❝♦♠♠✉t❛t✐✈❡s ❬✸✶( s✐♠✐•❛✐r❡s à ❝❡••❡s ❞❡s ❡s♣❛❝❡s L p ✳ ❊♥ ❡✛❡t✱ s♦✐t E ✉♥ ❡s♣❛❝❡ s②♠étr✐q✉❡ ❞❡ ❢♦♥❝t✐♦♥s s✉r (0, ∞)✱ ♥♦t♦♥s ♣❛r µ(x) •❛ ré❛rr❛♥❣é❡ ❞é❝r♦✐ss❛♥t❡ ❞✬✉♥ ♦♣ér❛t❡✉r x ✭✐✳❡✳ ✉♥❡ ❢♦♥❝t✐♦♥ s✉r (0, ∞) ❛②❛♥t •❛ ♠ê♠❡ ❞✐str✐❜✉t✐♦♥ q✉❡ |x|✮✳ ❖♥ ♣❡✉t ❛•♦rs ❞é✜♥✐r ✿ x E(M) = µ(x) E , ❡t ❛✐♥s✐ •✬❡s♣❛❝❡ E(M)✳ P❛r ❛❜✉s ❞❡ ♥♦t❛t✐♦♥✱ ♦♥ ♥♦t❡r❛ x E := x E(M) ✳ ❉é✜♥✐t✐♦♥ ✵✳✶✳✹ ✭◆♦r♠❡s •✐❣♥❡ ❡t ❝♦•♦♥♥❡✮✳ ❙♦✐t x ∈ S(M)✱ E ✉♥ ❡s♣❛❝❡ s②♠étr✐q✉❡✳ ❖♥ ❞é✜♥✐t •❡s ♥♦r♠❡s ❝♦•♦♥♥❡ ❡t •✐❣♥❡ ❝♦♠♠❡ s✉✐t ✿ x C E := i∈N x * i x i 1/2 E ❛♥❞ x R E := i∈N x i x * i 1/2 E ✳ ❯♥ éq✉✐✈❛•❡♥t ♣❡✉t t♦✉t ❞❡ ♠ê♠❡ êtr❡ tr♦✉✈é ❡♥ ❝♦♠❜✐♥❛♥t •❡s ❞❡✉① ❢♦♥❝t✐♦♥s ❝❛rrés✳ ❚❤é♦rè♠❡ ✵✳✶✳✼ ✭■♥é❣❛•✐tés ❞❡ ❑❤✐♥t❝❤✐♥❡ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡s✮✳ ❙♦✐t p ∈ [2, ∞)✳ ❆•♦rs ✿ i∈N x i ⊗ ξ i p ≈ max x Rp , x Cp . ❙♦✐t p ∈ (0, 2]✳ ❆•♦rs ✿ i∈N x i ⊗ ξ i p ≈ inf y+z=x z Cp + y Rp .
❈❡ rés✉•t❛t ❡st ❞✉ à ▲✉st✲P✐q✉❛r❞ ♣♦✉r p ∈ (1, ∞) ❬✸✻(✱ ▲✉st✲P✐q✉❛r❞ ❡t P✐s✐❡r ♣♦✉r p = 1 ❬✸✼( ❡t ♣•✉s ❞❡ ✈✐♥❣t ❛♥s ♣•✉s t❛r❞s✱ P✐s✐❡r ❡t |✐❝❛r❞ ❬✹✼( ♣♦✉r p < 1✳ P❛r •❛ ♠ét❤♦❞❡ ❞✬✐♥t❡r♣♦•❛t✐♦♥ ré❡••❡✱ •❡ rés✉•t❛t s✬ét❡♥❞ à •❛ ♣•✉♣❛rt ❞❡s ❡s♣❛❝❡s ❞❡ ▲♦r❡♥t③ q✉✐ s♦♥t ❞❡s ❡s♣❛❝❡s ❞❡ ❇❛♥❛❝❤ ♠❛✐s ♣❛s à L 2,∞ ♦✉ L 1,∞ ✳ P♦✉r L 2,∞ ✱ ❝❡•❛ ❡st ❞✉ ❛✉ ❢❛✐t q✉❡ ♣♦✉r é❝r✐r❡ L 2,∞ ❝♦♠♠❡ ✉♥ ❡s♣❛❝❡ ❞✬✐♥t❡r♣♦•❛t✐♦♥ ❞❡ L p ❡t L q ✱ ✐• ❡st ♥é❝❡ss❛✐r❡ q✉❡ p < 2 ❡t q > 3 ❡t ❞♦♥❝ ❞✬✉t✐•✐s❡r •❡s ❞❡✉① ❢♦r♠✉•❡s q✉✐ ♥❡ s✬✐♥t❡r♣♦•❡♥t ♣❛s ❜✐❡♥✳ ❉❡ ♠ê♠❡✱ ♣♦✉r é❝r✐r❡ L 1,∞ ❝♦♠♠❡ ✉♥ ❡s♣❛❝❡ ❞✬✐♥t❡r♣♦•❛t✐♦♥ ❞❡ L p ❡t L q ✱ ✐• ❡st ♥é❝❡ss❛✐r❡ q✉❡ p < 1 ❝❡ q✉✐ ❡st ♣r♦❜•é♠❛t✐q✉❡ ♣✉✐sq✉❡ •❛ ❢❡r♠❡t✉r❡ ❞❡ •✬❡s♣❛❝❡ ✈❡❝t♦r✐❡• ❣é♥éré ♣❛r •❡s ξ i ♥✬❡st ♣❛s ❝♦♠♣•é♠❡♥té ❞❛♥s L p (0, 1) ♣♦✉r p < 1✳ ❈❡s ❞❡✉① ❡①❡♠♣•❡s✱ q✉✐ s♦♥t ❞❡s ❡s♣❛❝❡s r❡

•❛t✐✈❡♠❡♥t ❝♦♠♠✉♥s✱ ❢✉r❡♥t •❛ ♠♦t✐✈❛t✐♦♥ ♦r✐❣✐♥❛•❡ ❞❡ ♥♦tr❡ tr❛✈❛✐•✳ ❙✉♣♣♦s♦♥s ♠❛✐♥t❡♥❛♥t q✉❡ •❡s ξ i s♦♥t ✉♥❡ s✉✐t❡ ❞❡ ✈❛r✐❛❜•❡s ❛•é❛t♦✐r❡s ♦rt❤♦♥♦r♠❛•❡s ❛r❜✐tr❛✐r❡s ❞❛♥s ✉♥ ❡s♣❛❝❡ ❞❡ ♣r♦❜❛❜✐•✐té ♥♦♥✲❝♦♠♠✉t❛t✐❢✳ ◆♦✉s ♦❜t❡♥♦♥s •❡ t❤é♦rè♠❡ s✉✐✈❛♥t ✿ ❚❤é♦rè♠❡ ✵✳✶✳✽✳ ❙♦✐t p < 1✳ ❙✉♣♣♦s♦♥s q✉❡ ♣♦✉r t♦✉t x ∈ S(M)✱ i≥0 x i ⊗ ξ i p inf y+z=x z Cp + y Rp . ❆•♦rs ♣♦✉r t♦✉t ❡s♣❛❝❡ ✐♥t❡r♣♦•é E ❡♥tr❡ L p ❡t L ∞ ❡t x ∈ S(M)✱ i≥ x i ⊗ ξ i E inf y+z=x z C E + y R E . ❉❡ ♣•✉s✱ •❛ ❞é❝♦♠♣♦s✐t✐♦♥ ❞❛♥s •❡ t❡r♠❡ ❞❡ ❣❛✉❝❤❡ ♣❡✉t êtr❡ ❝❤♦✐s✐❡ ✐♥❞é♣❡♥❞❛♥t❡ ❞❡ E✳
❈❡ t❤é♦rè♠❡ ✐♠♣•✐q✉❡ q✉❡ ♣♦✉r ❞❡s ✈❛r✐❛❜•❡s ❞❡ |❛❞❡♠❛❝❤❡r ✐♥❞é♣❡♥❞❛♥t❡s ♦✉ ❞❡s ✉♥✐t❛✐r❡s ❞❡ ❍❛❛r •✐❜r❡s✱ •❡s ✐♥é❣❛•✐tés ❞❡ ❑❤✐♥t❝❤✐♥❡ s♦♥t ✈ér✐✜é❡s ❞❛♥s L 1,∞ ✱ ♣✉✐sq✉❡✱ ❝♦♠♠❡ ♥♦✉s •✬❛✈♦♥t ♣ré❝é❞❡♠♠❡♥t ♠❡♥t✐♦♥♥é✱ •❡s ✐♥é❣❛•✐tés ❞❡ ❑❤✐♥t❝❤✐♥❡ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡s s♦♥t ✈r❛✐❡s ❞❛♥s L p ♣♦✉r p < 1 ♣♦✉r ❝❡s ✈❛r✐❛❜•❡s ✭♣❛r ❬✹✽(✮✳

❈♦r♦••❛✐r❡ ✵✳✶✳✾✳ ❙♦✐t (ξ i ) i∈N ✉♥❡ s✉✐t❡ ❞❡ ✈❛r✐❛❜•❡s ❞❡ |❛❞❡♠❛❝❤❡r ✐♥é♣❡♥✲ ❞❛♥t❡s ❡t x ∈ S(M)✳ ❆•♦rs ✿ i≥0 x i ⊗ ξ i 1,∞ ≈ inf y+z=x z C1,∞ + y R1,∞ .
▲❛ ♣r❡✉✈❡ ❞✉ t❤é♦rè♠❡ ❡st ❜❛sé❡ s✉r •✬✐❞é❡ ❞❡ ❝♦♥s✐❞ér❡r ❞❡s ❞é❝♦♠♣♦s✐t✐♦♥s ❖♥ ❞✐t q✉❡

x = y+z q✉✐ s♦♥t ♣r❡sq✉❡s ♦♣t✐♠❛•❡s ❞❛♥s •❡ s❡♥s ♦ù ❡••❡s ♠✐♥✐♠✐s❡♥t •❛ q✉❛♥t✐té y p Rp + z p Cp ✳ P♦✉r ❝❡s ❞é❝♦♠♣♦s✐t✐♦♥s✱ ✐• ♥✬❡st ♣❛s ❞✐✣❝✐•❡ ❞❡ ♠♦♥tr❡r✱ ❡♥ ✉t✐•✐s❛♥t •❡s ✐♥é❣❛•✐tés ❞❡ ❑❤✐♥t❝❤✐♥❡ ❞❛♥s L p ♣♦✉r ❞❡s s✉✐t❡s ❞❡ •❛ ❢♦r♠❡ (ex i ) i∈N ♦ù e ❡st ✉♥❡ ♣r♦•❡❝t✐♦♥ ❞❛♥s M✱ q✉❡ •❛ K✲❢♦♥❝t✐♦♥♥❡••❡ ❡♥ •✐❣♥❡ ❞❡ y ♣♦✉r •❡ ❝♦✉♣•❡ (L p , L ∞ ) ❡st ❞♦♠✐♥é❡ ♣❛r •❛ K✲❢♦♥❝t✐♦♥♥❡••❡ ❞❡ i∈N x i ⊗ ξ i ♣♦✉r •❡ ❝♦✉♣•❡ (L p , L ∞ )✳ ❉❡ ♠ê♠❡✱ ♣❛r ❞✉❛•✐té✱ ♦♥ ♠❛•♦r❡ •❛ K✲❢♦♥❝t✐♦♥♥❡••❡ ❡♥ ❝♦•♦♥♥❡ ❞❡ z ♣❛r •❛ K✲❢♦♥❝t✐♦♥♥❡••❡ ❞❡ i∈N x i ⊗ ξ i ✳ ❯♥❡ ❢♦✐s ✉♥❡ ♠❛•♦r❛t✐♦♥ ❞❡ •❛ K✲ ❢♦♥❝t✐♦♥♥❡••❡ ♦❜t❡♥✉❡✱ •❡ t❤é♦rè♠❡ ❞❡ ▲♦
x = (x i ) i≥0 ✉♥❡ s✉✐t❡ ✜♥✐❡ ❞✬é•é♠❡♥ts ❞❡ M c ✳ ■• ❡①✐st❡ α, β ∈ M + c ✱ ❡t u = (u i ) i≥0 ∈ M N c t❡•s q✉❡ ✿ • s(α) ≤ i≥0 u i u * i ≤ 1✱ • s(β) ≤ i≥0 u * i u i ≤ 1✱ • ♣♦✉r t♦✉t i ≥ 0✱ x i = u i β + αu i ✳ ❉❡ ♣•✉s✱ s✉♣♣♦s♦♥s q✉❡ ✿ • •❡s ξ i ❢♦r♠❡♥t ✉♥❡ s✉✐t❡ ❞❡ ✈❛r✐❛❜•❡s ♦rt❤♦♥♦r♠é❡s ❞❛♥s L 2 (A) • •❡s ξ i ✈ér✐✜❡♥t •✬✐♥é❣❛•✐té ❞❡ ❑❤✐♥t❝❤✐♥❡ ❞❛♥s L ∞ ✐✳❡ ♣♦✉r t♦✉t❡ s✉✐t❡ ✜♥✐❡ x = (x i ) i≥0 ∈ M N ✿ max x R∞ , x C∞ ≈ c∞ i≥0 x i ⊗ ξ i ∞ . ❆•♦rs ♣♦✉r t♦✉t p ∈ (0, ∞) ❡t E ✉♥ ❡s♣❛❝❡ ✐♥t❡r♣♦•é ❡♥tr❡ L p ❡t L ∞ ✿ α E + β E ≈ c i≥0 x i ⊗ ξ i E , ♦ù c ❞é♣❡♥❞ s❡✉•❡♠❡♥t ❞❡ p ❡t c ∞ ✳ ▲❛ ♣
■• ❡st ✐♠♣♦rt❛♥t ❞❡ r❡♠❛rq✉❡r q✉❡ •❡ t❤é♦rè♠❡ ❞♦♥♥❡ ✉♥ éq✉✐✈❛•❡♥t ❞ét❡r♠✐♥✐st❡ ❞❡ •❛ ♥♦r♠❡ ❞❡ i≥0 x i ⊗ ξ i ❞❛♥s L 2,∞ ❡t ré♣♦♥❞ ❛✐♥s✐ ❞✬✉♥❡ ❝❡rt❛✐♥❡ ❢❛ç♦♥ à ✉♥❡ q✉❡st✐♦♥ ♣♦sé❡ ♣❛r ▲✉st✲P✐q✉❛r❞ ❡t ❳✉ ❞❛♥s ❬✸✽(✳ ❈❡t éq✉✐✈❛•❡♥t s❡♠❜•❡ ❞✐✣❝✐•❡ à ♠❛♥✐♣✉•❡r ❡♥ ✈✉❡ ❞✬❛♣♣•✐❝❛t✐♦♥s ♠❛✐s ✐• ❛ •✬❛✈❛♥t❛❣❡ ❞❡ ♣r❡♥❞r❡ •❛ ♠ê♠❡ ❢♦r♠❡ ❞❛♥s t♦
♦rè♠❡ ✵✳✶✳✶✶✳ ❙♦✐t E ✉♥ ❡s♣❛❝❡ s②♠♠étr✐q✉❡ ❛②❛♥t •❛ ♣r♦♣r✐été ❞❡ ❋❛t♦✉ ❡t M = B(ℓ 2 )⊗L ∞ (0, 1)✳ ❆•♦rs E ❡st 2✲♠♦♥♦t♦♥❡ à ❣❛✉❝❤❡ s✐ ❡t s❡✉•❡♠❡♥t s✐ ♣♦✉r t♦✉t❡ s✉✐t❡ ✜♥✐❡ x ∈ M N c ✱ x R E ∩C E ≈ i≥0 x i ⊗ ξ i E ✳ ❙✉♣♣♦s♦♥s ❞❡ ♣•✉s q✉✬✐• ❡①✐st❡ p > 0 t❡• q✉❡ E ❡st ✉♥ ❡s♣❛❝❡ ✐♥t❡r♣♦•é ❡♥tr❡ L p ❡t L ∞ ✳ ❆•♦rs E ❡st 2✲♠♦♥♦t♦♥❡ à ❞r♦✐t❡ s✐ ❡t s❡✉•❡♠❡♥t s✐ ♣♦✉r t♦✉t❡ s✉✐t❡ ✜♥✐❡ x ∈ M N c ✱ x R E +C E ≈ i≥0 x i ⊗ ξ i E ✳ ❈❡❝✐ ✈✐❡♥t ❡♥ ❝♦♠♣•é♠❡♥t à •❛ ❢♦r♠✉•❡ ❞ét❡r♠✐♥✐st❡ ❞♦♥♥é❡ ❞❛♥s •❡ t❤é♦rè♠❡ ♣ré❝é❞❡♥t✱ ♠♦♥tr❛♥t ♣❛r ❡①❡♠♣•❡ q✉❡ ❞❛♥s •❡ ❝❛s ❞❡ E = L 2,∞ ✱ ❛✉❝✉♥❡ ❞❡s ❞❡✉① ❢♦r♠✉•❡s ✉s✉❡••❡s ♥❡ ❢♦♥❝t✐♦♥♥❡♥t✳ ▲❛ ♣r❡✉✈❡ ❞✉ t❤é♦rè♠❡ s❡ ❜❛s❡ ❡ss❡♥t✐❡••❡✲ ♠❡♥t s✉r •✬✉t✐•✐s❛t✐♦♥ ❞✉ t❤é♦rè♠❡ ❞❡ ❙❝❤✉r✲❍♦r♥✳ ■♥é❣❛•✐tés ❞❡ t②♣❡ ❢❛✐❜•❡ ♣♦✉r •❡s ✐♥té❣r❛•❡s s✐♥❣✉•✐èr❡s ❉❛♥s •❡ ❞❡r♥✐❡r ❝❤❛♣îtr❡✱ ♥♦✉s ♣rés❡♥t♦♥s •❡ tr❛✈❛✐• q✉✐ ❛♣♣❛r❛ît ❞❛♥s •✬❛rt✐❝•❡ ❬✺(✳ ■• ❡st ♣r❡sq✉❡ ✐♥❞é♣❡♥❞❛♥t ❞✉ r❡st❡ ❞❡ •❛ t❤ès❡✱ s✐ ❝❡ ♥✬❡st ♣♦✉r •❡s ❞é✜♥✐t✐♦♥s ❞❡ ❜❛s❡ ❡t ✉♥❡ ♣❡t✐t❡ ❛♣♣•✐❝❛t✐♦♥ ❞❡s ✐♥é❣❛•✐tés ❞❡ ❑❤✐♥t❝❤✐♥❡ ❞❛♥s L 1,∞ ❞♦♥♥é❡ à •❛ ✜♥ ❞✉ ❝❤❛♣îtr❡✳ ■• tr❛✐t❡ ❞❡s ♦♣ér❛t❡✉rs ❞❡ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦✉ ✐♥té❣r❛•❡s s✐♥❣✉•✐èr❡s ✐✳❡✳ ✉♥ ❝❡rt❛✐♥ t②♣❡ ❞✬♦♣ér❛t❡✉rs T ❞é✜♥✐s ❞❡ ❢❛ç♦♥ ✐♥❢♦r♠❡••❡ ♣❛r ❞❡s ❡①♣r❡ss✐♦♥s ❞❡ •❛ ❢♦r♠❡ ✿ T f (x) = R n k(x, y)f (y)dy, ♦ù n ≥ 1✱ x ∈ R n
♥ ✐♥❢♦r♠❡••❡✳ ❚❤❡♦r❡♠ ✵✳✶✳✶✷ ✭❈❛•❞❡ró♥✱ )②❣♠✉♥❞✮✳ ❙✐ T ❡st ❜♦r♥é s✉r L 2 (R n ) ❡t ❡st ❛ss♦✲ ❝✐é à ✉♥ ♥♦②❛✉ k ✈ér✐✜❛♥t ❞❡s ❝♦♥❞✐t✐♦♥s ❞❡ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ❛•♦rs T s✬ét❡♥❞ ❡♥ ✉♥ ♦♣ér❛t❡✉r ❜♦r♥é ❞❡ L 1 (R n ) ✈❡rs L 1,∞ (R n )✳ ❈❡ t❤é♦rè♠❡ ✐♠♣•✐q✉❡ ♣❛r ✐♥t❡r♣♦•❛t✐♦♥ ✲ ♣•✉s ♣ré❝✐sé♠❡♥t ♣❛r •❡ t❤é♦rè♠❡ ❞❡ ▼❛r❝✐♥❦✐❡✇✐❝③ ❝✐té ♣•✉s ❤❛✉t ✲ ❡t ♣❛r ❞✉❛•✐té q✉❡ •❡s ♦♣ér❛t❡✉rs ❞❡ ❈❛•❞❡ró♥✲ )②❣♠✉♥❞ s♦♥t ❜♦r♥és ❞❡ L p (R n ) ✈❡rs L p (R n ) ♣♦✉r t♦✉t p ∈ (1, ∞)✳ P❛r •❛ ♠ê♠❡ ❡①♣r❡ss✐♦♥✱ T ♣♦✉rr❛✐t ❞é✜♥✐r ✉♥ ♦♣ér❛t❡✉r ❞❡ L p (R n , X) ✈❡rs L p (R n , X)
♣♣•✐❝❛t✐♦♥s✱ ♣❛r♠✐ •❡sq✉❡•s ✉♥❡ ❡♥ t❤é♦r✐❡ ❞❡ •❛ ♣❡rt✉r❜❛t✐♦♥✳ ▲❡s ✐♥t❡r❛❝t✐♦♥s ❡♥tr❡ •❛ t❤é♦r✐❡ ❞❡ •❛ ♣❡rt✉r❜❛t✐♦♥ ❡t •✬❛♥❛②s❡ ❤❛r♠♦♥✐q✉❡ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ♦♥t ❞é❜✉té •♦rsq✉❡ P♦t❛♣♦✈ ❡t ❙✉❦♦❝❤❡✈ ♦♥t rés♦•✉ ✉♥ ♣r♦❜✲ •è♠❡ ♣♦sé ♣❛r ❑r❡✐♥ ❡t r❡sté •♦♥❣t❡♠♣s ✐♥t♦✉❝❤é s✉r •❡ ❝❛•❝✉• ❢♦♥❝t✐♦♥♥❡• ▲✐♣s✲ ❝❤✐t③✐❡♥ ❞❛♥s •❡s ❝•❛ss❡s ❞❡ ❙❝❤❛tt❡♥ ❬✺✶(✳ ■•s ♦♥t ♣r♦✉✈é q✉❡ ♣♦✉r t♦✉t p ❞❛♥s (1, ∞)✱ ✐• ❡①✐st❡ ✉♥❡ ❝♦♥st❛♥t❡ c p t❡••❡ q✉❡ ♣♦✉r t♦✉t❡ ❢♦♥❝t✐♦♥ ▲✐♣s❝❤✐t③✐❡♥♥❡ f : R → R ❡t t♦✉s ♦♣ér❛t❡✉rs ❛✉t♦✲❛❞•♦✐♥ts x ❡t y✱ f (x) -f (y) p ≤ c p f Lip x -y p , ✭✷✮ ♦ù •✬♦♥ s✉♣♣♦s❡ q✉❡ x -y ∈ L p (M) ❡t ♦♥ ❞é✜♥✐t ✿
T ✈ér✐✜❡ •✬❤②♣♦t❤ès❡ ❞❡ t❛✐••❡ s✐ ♣♦✉r t♦✉t x, y ∈ R n ✱ k(x, y) M 1 |x -y| n . ❚❤é♦rè♠❡ ✵✳✶✳✶✸✳ ❖♥ s✉♣♣♦s❡ q✉❡ T ❛ ♣♦✉r ♣❛r❛♠ètr❡ ❞❡ ▲✐♣s❝❤✐t③ γ ✭0 < γ ≤ 1✮✱ ✈ér✐✜❡ •✬❤②♣♦t❤ès❡ ❞❡ t❛✐••❡ ❡t ❡st L 2 (R n ) ❜♦r♥é s✉r L 2 (R n )✳ ❆•♦rs T s✬ét❡♥❞ ❡♥ ✉♥ ♦♣ér❛t❡✉r ❜♦r♥é ❞❡ L 1 (M⊗L ∞ (R n )) ✈❡rs L 1,∞ (M⊗L ∞ (R n ))✳ ▲❛ ♣r❡✉✈❡ ❞❡ ❝❡ t❤é♦rè♠❡✱ ❞❛♥s •❡ ❝❛s ❝•❛ss✐q✉❡✱ r❡♣♦s❡ s✉r ✉♥❡ ❞é❝♦♠♣♦s✐t✐♦♥ ❞❡ ❢♦♥❝t✐♦♥ f = g + b✱ q✉✐ ♣❡✉t êtr❡ ✐♥t❡r♣rété❡ ❞❛♥s •❡ •❛♥❣✉❛❣❡ ❞❡s ♠❛rt✐♥✲ ❣❛•❡s✳ ❈❡tt❡ ✐♥t❡r♣rét❛t✐♦♥ ♣❡r♠❡t ❞❡ ❝♦♥str✉✐r❡ ✉♥❡ ✈❡rs✐♦♥ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❞❡ •❛ ❞é❝♦♠♣♦s✐t✐♦♥✱ ❡♥ ✉t✐•✐s❛♥t •❛ t❤é♦r✐❡ ❜✐❡♥ ❞é✈❡•♦♣♣é❡ ❞❡s ♠❛rt✐♥❣❛•❡s ♥♦♥✲❝♦♠♠✉t❛t✐✈❡s ❡t ❡♥ ♣❛rt✐❝✉•✐❡r •❡s ♣r♦•❡❝t✐♦♥s ❞❡ ❈✉❝✉•❡s❝✉✳ ❈❡♣❡♥❞❛♥t✱ •❛ ❞é❝♦♠♣♦s✐t✐♦♥ ♦❜t❡♥✉❡ ❞❛♥s •❡ ❝❛❞r❡ ♥♦♥✲❝♦♠♠✉t❛t✐❢ ♥❡ s❛t✐s❢❛✐t ♣❛s t♦✉t❡s •❡s ❜♦♥♥❡s ♣r♦♣r✐étés ❞❡ •❛ ❞é❝♦♠♣♦s✐t✐♦♥ ❝•❛ss✐q✉❡✳ ❉❡ ♥♦✉✈❡❛✉① t❡r♠❡s ❛♣♣❛✲ r❛✐ss❡♥t ❡t ♣♦✉r •❡s ♠❛•♦r❡r✱ •❡ t❤é♦rè♠❡ ❞❡ ♣s❡✉❞♦•♦❝❛•✐s❛t✐♦♥ L 2 s✉✐✈❛♥t ❡st ♥é❝❡ss❛✐r❡✳ ❖♥ ❞é✜♥✐t N = M⊗L ∞ (R n ) ❡t N k •❛ s♦✉s✲❛•❣è❜r❡ ❞❡ N ❝♦♠♣♦sé❡ ❞❡s ❢♦♥❝t✐♦♥s ❝♦♥st❛♥t❡s s✉r •❡s ❝✉❜❡s ❞②❛❞✐q✉❡s ❞❡ ❝ôté ❞❡ •♦♥❣✉❡✉r 2 -k ✳ ❚❤é♦rè♠❡ ✵✳✶✳✶✹ ✭Ps❡✉❞♦•♦❝❛•✐s❛t✐♦♥ ❞❛♥s L 2 ✮✳ ❙♦✐t f ∈ L 2 (N ) ❛♥❞ s ∈ N✳ P♦✉r t♦✉t k ∈ Z✱ s♦✐❡♥t A k ❡t B k ❞❡s ♣r♦•❡❝t✐♦♥s ❞❛♥s N k t❡••❡s q✉❡ A ⊥ k df k+s = df k+s B ⊥ k = 0✳ P♦✉r t♦✉t ❡♥t✐❡r ♣♦s✐t✐❢ ✐♠♣❛✐r d✱ ♦♥ ❞é✜♥✐t ✿ A k = Q∈Q k A Q I Q ✱ A Q ∈ M ❛♥❞ ❞❡✜♥❡ dA k := Q∈Q k A Q I dQ ✳ ❉❡ ♠ê♠❡✱ ♦♥ ❞é✜♥✐t dB k ✳ ❙♦✐t ✿ A f,s := k∈Z 5A k ❛♥❞ B f,s := k∈Z 5B k . ❙♦✐t T ✉♥ ♦♣ér❛t❡✉r ❛ss♦❝✐é à ✉♥ ♥♦②❛✉ k ❞❡ ♣❛r❛♠ètr❡ ❞❡ ▲✐♣s❝❤✐t③ γ✱ ✈ér✐✜❛♥t •❛ ❝♦♥❞✐t✐♦♥ ❞❡ t❛✐••❡✱ ❡t ❜♦r♥é ❞❡ L 2 (N ) ✈❡rs L 2 (N )✳ ❆•♦rs ♣♦✉r t♦✉t s ∈ N ❡t f ∈ L 2 (N ) ♦♥ ❛ ✿
❚❤✐s ♠❡❛♥s t❤❡ s❡t ♦❢ ✈❛•✉❡s p s✉❝❤ t❤❛t T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L p (Ω) t♦ L p (Ω) ✐s ❛♥ ✐♥t❡r✈❛•✳ ❖♥❡ ♠♦t✐✈❛t✐♦♥ ❢♦r t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ✈❛r✐♦✉s s♣❛❝❡s ✐♥ ❤❛r♠♦♥✐❝ ❛♥❛•②s✐s ✐s t♦ st✉❞② t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ T ❛t t❤❡ ❡♥❞♣♦✐♥ts ♦❢ t❤✐s ✐♥t❡r✈❛

•✳ ■❢ T ✐s ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r ❢♦r ❡①❛♠♣•❡ ♦r ♠♦r❡ ♣❛rt✐❝✉•❛r•② t❤❡ ❍✐•❜❡rt tr❛♥s❢♦r♠ t❤❡♥ T ✐s ❜♦✉♥❞❡❞ ♦♥ L p (R) ❢♦r p ∈ (1, ∞) ❜✉t ♥♦t ❢♦r p = 1 ❛♥❞ p = ∞✳ ❍♦✇❡✈❡r✱ T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L 1 t♦ L 1,∞ ✭✇❡ s❛② t❤❛t T ✐s ♦❢ ✇❡❛❦ t②♣❡ (1, 1)✮ ❛♥❞ ❢r♦♠ L ∞ t♦ BM O✳ ❑♥♦✇✐♥❣ s✉❝❤ ❡♥❞♣♦✐♥t ✐♥❡q✉❛•✐t✐❡s ✐s t❤❡♥ tr❛♥s•❛t❡❞ ❜② ✐♥t❡r♣♦•❛t✐♦♥ t❤❡♦r② ✐♥t♦ ✐♥❡q✉❛•✐t✐❡s ♦♥ L p ✳ ❚❤❡ ♣r✐♠❡ ❡①❛♠♣•❡ ♦❢ s✉❝❤ ✐♥t❡r♣♦•❛t✐♦♥ t❤❡♦r❡♠ ✐s t❤❡ ❢♦••♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✵✳✷✳✷ ✭▼❛r❝✐♥❦✐❡✇✐❝③✮✳ ▲❡t p < q ∈ (0, ∞)✳ ■❢ T ✐s ♦❢ ✇❡❛❦ t②♣❡ (p, p) ❛♥❞ ♦❢ ✇❡❛❦ t②♣❡ (q, q) t❤❡♥ T ✐s ❜♦✉♥❞❡❞ ♦♥ L r (Ω)✱ r ∈ (p, q)✳ ❚❤✐s ♠❡❛♥s t❤❛t ✇❡❛❦ t②♣❡ ✐♥❡q✉❛•✐t✐❡s ❝❛♥ ❜❡ ❡♥♦✉❣❤ t♦ ♣r♦✈❡ t❤❡ ❜♦✉♥❞✲ ❡❞♥❡ss ♦❢ ❝❡rt❛✐♥ ♦♣❡r❛t♦rs✳ ❚❤✐s ✐s ❢♦r ❡①❛♠♣•❡ t❤❡ ✇❛② t❤❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ s✐♥❣✉•❛r ✐♥t❡❣r❛•s ✇❛s ♦r✐❣✐♥❛••② ♣r♦✈❡❞ ❜② ❈❛•❞❡ró♥ ❛♥❞ )②❣♠✉♥❞✳ ❆♥♦t❤❡r ♠♦t✐✈❛t✐♦♥ ❢♦r ♦❜t❛✐♥✐♥❣ ❡♥❞♣♦✐♥t ❡st✐♠❛t❡s ❛t ❛ ♣♦✐♥t p ✐s t❤❛t t❤❡② ♦❢t❡♥ ❣✐✈❡ ❛♥ ♦♣t✐♠❛• ❝♦♥tr♦• ♦♥ t❤❡ ♥♦r♠ ♦❢ T ♦♥ L q ✇❤❡♥ q ❣♦❡s t♦ p✳ ❆❜str❛❝t•②✱ t❤❡ t❤❡♦r② ♦❢ ✐♥t❡r♣♦•❛t✐♦♥ ❞❡❛•s ✇✐t❤ ❝♦✉♣•❡s (A, B) ♦❢ q✉❛s✐✲ ❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞✱ ✐♥❢♦r♠❛••②✱ ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ A ❛♥❞ B ✐s ❛ q✉❛s✐✲❇❛♥❛❝❤ s♣❛❝❡ E s✉❝❤ t❤❛t ✐❢ ❛♥ ♦♣❡r❛t♦r T ✐s ❜♦✉♥❞❡❞ ♦♥ A ❛♥❞ B✱ ✐t ✐s ❛✉t♦♠❛t✐❝❛••② ❜♦✉♥❞❡❞ ♦♥ E✳ ❚♦ ❛♣♣•② t❤❡ ♠❛❝❤✐♥❡r② ♦❢ ✐♥t❡r♣♦•❛t✐♦♥✱ ✐t ✐s t❤❡r❡❢♦r❡ ✉s❡❢✉• t♦ ❦♥♦✇ ❤♦✇ t♦ ❞❡s❝r✐❜❡ t❤❡ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ♦❢ ❛ ❣✐✈❡♥ ❝♦✉♣•❡✳ ❚❤❡r❡ ✐s ♥♦ ❡✛❡❝t✐✈❡ ✇❛② t♦ ❛❝❝♦♠♣•✐s❤ t❤✐s ✐♥ ❢✉•• ❣❡♥❡r❛•✐t② ❜✉t ✐t ❝❛♥ ❜❡ ❞♦♥❡ ❢♦r t❤❡ ❝♦✉♣•❡ L p (0, ∞)✱ L q (0, ∞) ✇✐t❤ p, q ∈ (0, ∞] ❜② ✉s✐♥❣ t❤❡ ♥♦t✐♦♥ ♦❢ K✲❢✉♥❝t✐♦♥❛• ❛♥❞ ✐♥ ♣❛rt✐❝✉•❛r K✲♠♦♥♦t♦♥✐❝✐t② ✭✶✳✷✳✹✮✳ ❋♦r t❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤✐s t❡①t✱ ❢♦r t✇♦ q✉❛♥t✐t✐❡s A(x) ❛♥❞ B(x)✱ ✇❡ ✇✐•• ✇r✐t❡ A(x) B(x) ♦r A(x) C B(x)✱ ✐❢ t❤❡r❡ ✐s ❛ ✉♥✐✈❡rs❛• ❝♦♥st❛♥t C s✉❝❤ t❤❛t ❢♦r ❛•• x✱ A(x) ≤ CB(x)✱ ❛♥❞ s✐♠✐•❛r•② A(x) ≈ B(x) ♦r A(x) ≈ C B(x) ✐❢ t❤❡r❡ ❡①✐sts C s✉❝❤ t❤❛t ❢♦r ❛•• x✱ 1 C B(x) ≤ A(x) ≤ CB(x)✳ |♦✉❣❤•②✱ ✇❡ s❛② t❤❛t ❛ s♣❛❝❡ E ⊂ A + B ✐s K✲♠♦♥♦t♦♥❡ ❢♦r ❛ ❝♦✉♣•❡ (A, B) ✐❢ ❢♦r ❛•• x, y ∈ E✱ x ≻ A,B y ✐♠♣•✐❡s t❤❛t x E y E ✱ ✇❤❡r❡ ≻ A,B ✐s ❛♥ ♦r❞❡r ❞❡♣❡♥❞✐♥❣ ♦♥ (A, B)✳ ■t ✐s ❛ ❜❛s✐❝ ❢❛❝t t❤❛t ❢♦r ❛♥② ❝♦✉♣•❡✱ ❡✈❡r② K✲♠♦♥♦t♦♥❡ s♣❛❝❡ ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡✳ ■♥ ❝❡rt❛✐♥ ❝❛s❡s✱ t❤❡ ❝♦♥✈❡rs❡ ✐s ❛•s♦ tr✉❡✳ ❚❤❡♦r❡♠ ✵✳✷✳✸✳ ❊✈❡r② ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ❝♦✉♣•❡s ♦❢ q✉❛s✐✲ ❇❛♥❛❝❤ s♣❛❝❡s ✐s K✲♠♦♥♦t♦♥❡✿ • ✭▲♦r❡♥t③✲❙❤✐♠♦❣❛❦✐✮ p ≥ 1✱ (L p (0, α), L ∞ (0, α)) ❛♥❞ (L 1 (0, α), L p (0, α))✱ • ✭❙♣❛rr✮ p, q ≥ 1✱ (L p (Ω), L q (Ω)) ❢♦r ❛♥② σ✲✜♥✐t❡ ♠❡❛s✉r❡ s♣❛❝❡ Ω✳ • ✭❙♣❛rr✱ ❈✇✐❦❡•✮ p, q ∈ (0, ∞)✱ (L p (0, α), L q (0, α))✱ • ✭❈✇✐❦❡•✮ p ∈ (0, ∞]✱ (ℓ p , ℓ ∞ )✳ ❲❡ ❡①t❡♥❞ ▲♦r❡♥t③ ❛♥❞ ❙❤✐♠♦❣❛❦✐✬s r❡s✉•t t♦ t❤❡ q✉❛s✐✲❇❛♥❛❝❤ s❡tt✐♥❣ ✇✐t❤ t❤❡ ❢♦••♦✇✐♥❣ t❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✵✳✷✳✹✳ ▲❡t α ∈ (0, ∞] ❛♥❞ p ∈ (0, ∞)✳ ❚❤❡♥ ❡✈❡r② ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ♦❢ t❤❡ ❝♦✉♣•❡ (L p (0, α), L ∞ (0, α)) ✐s K✲♠♦♥♦t♦♥❡✳ ❚❤❡ ♣r♦♦❢ ❞♦❡s ♥♦t ♣r❡s❡♥t ❛♥② ♠❛•♦r ❞✐✣❝✉•t②✱ ✐t ❝♦♥s✐sts✱ ❢♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f, g s✉❝❤ t❤❛t f ≻ Lp,L∞ g✱ ✐♥ ❡①♣•✐❝✐t❡•② ❝♦♥str✉❝t✐♥❣ ❛♥ ♦♣❡r❛t♦r T ❜♦✉♥❞❡❞ ♦♥ L p ❛♥❞ L ∞ s✉❝❤ t❤❛t T f = g✳ ◆♦t❡ t❤❛t ✐t ✐s ❛•s♦ ❦♥♦✇♥ ❜② ❙♣❛rr ❬✺✼(✱ t❤❛t ❡✈❡r② ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ♦❢ t❤❡ ❝♦✉♣•❡ (L p (0, α), L q (0, α)) ✐s K✲♠♦♥♦t♦♥❡ ❢♦r p, q ∈ (0, ∞)✳ ❲❡ ❣✐✈❡ ❛♥ ❛•t❡r♥❛t✐✈❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤♦s❡ s♣❛❝❡s ✐♥ t❡r♠s ♦❢ t✇♦ ♥♦t✐♦♥s ♦❢ ♠♦♥♦t♦♥✐❝✐t②✳ ❉❡✜♥✐t✐♦♥ ✵✳✷✳✺ ✭p✲♠♦♥♦t♦♥✐❝✐t②✮✳ ▲❡t E ❜❡ ❛ ❢✉♥❝t✐♦♥ s♣❛❝❡ ♦♥ (0, ∞)✱ ✇❡ s❛② t❤❛t E ✐s •❡❢t✲p✲♠♦♥♦t♦♥❡ ✐❢ ❢♦r ❛•• ❢✉♥❝t✐♦♥s f ∈ E ❛♥❞ g ∈ L 0 (0, ∞) s✉❝❤ t❤❛t ❢♦r ❛•• t > 0✱ t 0 (f * ) p ≥ t 0 (g * ) p t❤❡♥ g ∈ E ❛♥❞ g E f E ✳ ❙✐♠✐•❛r•②✱ ✇❡ s❛② t❤❛t E ✐s r✐❣❤t✲p✲♠♦♥♦t♦♥❡ ✐❢ ❢♦r ❛•• ❢✉♥❝t✐♦♥s f ∈ E ❛♥❞ g ∈ L 0 (0, ∞) s✉❝❤ t❤❛t ❢♦r ❛•• t > 0✱ ∞ t (f * ) p ≥ ∞ t (g * ) p t❤❡♥ g ∈ E ❛♥❞ g E f E ✳ ❲❡ ♣r♦✈❡ t❤❡ ❢♦••♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✵✳✷✳✻✳ ▲❡t E ❜❡ ❛ ❢✉♥❝t✐♦♥ s♣❛❝❡ ♦♥ (0, α)✱ α > 0✱ ❛♥❞ •❡t 0 < p < q < ∞✳ ❚❤❡♥ E ✐s •❡❢t✲p✲♠♦♥♦t♦♥❡ ❛♥❞ r✐❣❤t✲q✲♠♦♥♦t♦♥❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ L p (0, α) ❛♥❞ L q (0, α)✳ ❖✉r ♣r♦♦❢ r❡•✐❡s ♦♥ ❙♣❛rr✬s r❡s✉•t✳ ❲❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ ♣r♦✈✐♥❣ t❤❛t ❛ s♣❛❝❡ E ✐s •❡❢t✲p✲♠♦♥♦t♦♥❡ ❛♥❞ r✐❣❤t✲q✲♠♦♥♦t♦♥❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ ✐t ✐s K✲♠♦♥♦t♦♥❡ ❢♦r t❤❡ ❝♦✉♣•❡ (L p , L q )✳ ❲❡ ❛•s♦ s❤♦✇ t❤❛t ❢♦r ❛ ❢✉♥❝t✐♦♥ s♣❛❝❡ ✇✐t❤ t❤❡ ❋❛t♦✉ ♣r♦♣❡rt②✱ p✲❝♦♥✈❡①✐t② ✐♠♣•✐❡s •❡❢t✲p✲♠♦♥♦t♦♥❝✐t② ❛♥❞ s✐♠✐•❛r•② q✲❝♦♥❝❛✈✐t② ✐♠♣•✐❡s r✐❣❤t✲q✲ ♠♦♥♦t♦♥✐❝✐t②✳ |❡❝❛•• t❤❛t ❛ s♣❛❝❡ E ✐s s❛✐❞ t♦ ❜❡ ♥❛t✉r❛• ✐❢ t❤❡r❡ ❡①✐sts ❛ p > 0 s✉❝❤ t❤❛t E ✐s p✲❝♦♥✈❡①✳ ❚❤❡♥ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦••♦✇✐♥❣ ❝♦r♦••❛r②✳ ❈♦r♦••❛r② ✵✳✷✳✼✳ ▲❡t E ❜❡ ❛ ♥❛t✉r❛• ❢✉♥❝t✐♦♥ s♣❛❝❡ ♦♥ (0, α) ✇✐t❤ t❤❡ ❋❛t♦✉ ♣r♦♣❡rt②✳ ❚❤❡♥ E ✐s r✐❣❤t✲q✲♠♦♥♦t♦♥❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ t❤❡r❡ ❡①✐sts 0 < p < q s✉❝❤ t❤❛t E ∈ (L p (0, α), L q (0, α))✳ ❚❤✐s s♦•✈❡s✱ ✉♥❞❡r ❛ss✉♠♣t✐♦♥s ✇❤✐❝❤ ❛r❡ ♠✐♥♦r ✐♥ ♣r❛❝t✐❝❡✱ ❛ ❝♦♥•❡❝t✉r❡ ♦❢ ▲❡✈✐t✐♥❛✱ ❙✉❦♦❝❤❡✈ ❛♥❞ )❛♥✐♥ ❬✸✹( ✐♥ t❤❡ s❡tt✐♥❣ ♦❢ ❢✉♥❝t✐♦♥ s♣❛❝❡s✳ ◆♦t❡ t❤❡ K✲♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ ❝♦✉♣•❡ (ℓ p , ℓ q ) ✐s ♥♦t ❦♥♦✇♥ ❛♥❞ t❤❡ ❝♦♥•❡❝t✉r❡ r❡♠❛✐♥s ❢✉••② ♦♣❡♥ ❢♦r s❡q✉❡♥❝❡ s♣❛❝❡s✳ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ s②♠♠❡tr✐❝ s♣❛❝❡s ❚❤❡ ♦r✐❣✐♥❛• ♠♦t✐✈❛t✐♦♥ ❢♦r t❤✐s ✇♦r❦ ♦♥ ❛❜str❛❝t ✐♥t❡r♣♦•❛t✐♦♥ t❤❡♦r② ❝♦♠❡s ❢r♦♠ t❤❡ st✉❞② ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s✳ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ❛r❡ ❡ss❡♥t✐❛• t♦ st✉❞② ✉♥❝♦♥❞✐t✐♦♥❛• s❡q✉❡♥❝❡s ✐♥ ♥♦♥✲ ❝♦♠♠✉t❛t✐✈❡ L p ✲s♣❛❝❡s✳ ❚❤❡② ♣•❛②❡❞ ❛♥ ✐♠♣♦rt❛♥t r♦•❡ ✐♥ t❤❡ ❞❡✈❡•♦♣♠❡♥t ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❤❛r♠♦♥✐❝ ❛♥❛•②s✐s ♦✈❡r t❤❡ •❛st t❤r❡❡ ❞❡❝❛❞❡s✱ ❜♦t❤ ❛s ❛ t♦♦• ❛♥❞ ❜② ♣r♦✈✐❞✐♥❣ t❤❡ ✧r✐❣❤t✧ ❞❡✜♥✐t✐♦♥ ❢♦r sq✉❛r❡ ❢✉♥❝t✐♦♥s ✐♥ t❤✐s ❝♦♥t❡①t ✇❤✐❝❤ ✐s ❛t t❤❡ ♦r✐❣✐♥ ♦❢ ♠❛•♦r ❞❡✈❡•♦♣♠❡♥ts s✉❝❤ ❛s ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❛rt✐♥❣❛•❡ ✐♥✲ ❡q✉❛•✐t✐❡s ❬✹✾(✳ ▲❡t ✉s st❛rt ❜② r❡❝❛••✐♥❣ t❤❡ st❛t❡♠❡♥t ♦❢ ❝•❛ss✐❝❛• ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s✳ ▲❡t (ξ i ) i∈N ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ |❛❞❡♠❛❝❤❡r ✈❛r✐❛❜•❡s ✐♥ L ∞ (0, 1) ❛♥❞ (a i ) i∈N ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣•❡① ♥✉♠❜❡rs✳ ❚❤❡♥ ❢♦r ❛•• p ∈ (0, ∞)✿ i≥0 a i ξ i p ≈ cp i≥0 a i ξ i 2 . ❚❤✐s ❡q✉✐✈❛•❡♥❝❡ ❡①t❡♥❞s ✐♠♠❡❞✐❛t❡•② t♦ s✉♠s ✇❤♦s❡ ❝♦❡✣❝✐❡♥ts t❛❦❡ ✈❛•✉❡s ✐♥ ❝•❛ss✐❝❛• L p ✲s♣❛❝❡s✳ ▲❡t Ω ❜❡ ❛ ♠❡❛s✉r❡ s♣❛❝❡✱ p ∈ (0, ∞) ❛♥❞ (f i ) i∈N ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ✐♥ L p (Ω)✳ ❚❤❡♥✿ i≥0 f i ξ i Lp(Ω⊗(0,1)) ≈ cp i≥0 |f i | 2 1/2 Lp(Ω) . ✭✸✮ ❯♥❞❡r t❤✐s s❡❝♦♥❞ ❢♦r♠✱ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ❛♣♣❡❛r ❢r❡q✉❡♥t•② ✐♥ ❤❛r♠♦♥✐❝ ❛♥❛•②s✐s t♦ ❡st✐♠❛t❡ sq✉❛r❡ ❢✉♥❝t✐♦♥s ♦r st✉❞② t❤❡ R✲❜♦✉♥❞❡❞♥❡ss ♦❢ ❢❛♠✐•✐❡s ♦❢ ❢✉♥❝t✐♦♥s✱ ❛ ♣r♦❜•❡♠ ✇❤✐❝❤ ✐s ❡ss❡♥t✐❛• ✇❤❡♥ tr②✐♥❣ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❢✉♥❝t✐♦♥❛• ❝❛•❝✉•✉s ♦♥ L p ✲s♣❛❝❡s✳ ◆♦t❡ t❤❛t t❤❡ ❡①♣r❡ss✐♦♥ ♦♥ t❤❡ •❡❢t ❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❡q✉✐✈❛•❡♥❝❡ ❝❛♥ ✐♥ ❢❛❝t ❜❡ ❞❡✜♥❡❞ ❢♦r ❛♥② ❇❛♥❛❝❤ s♣❛❝❡ X ✐♥st❡❛❞ ♦❢ L p (Ω)✱ ❜② s❡❡✐♥❣ i≥0 f i ξ i ❛s ❛♥ ❡•❡♠❡♥t ♦❢ L p ((0, 1), X)✱ ❛♥❞ ❛ ✈❡rs✐♦♥ ♦❢ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ❝❛••❡❞ ❑❤✐♥t❝❤✐♥❡✲❑❛❤❛♥❡ ✐♥❡q✉❛•✐t② ❤♦•❞s ❛t t❤❛t •❡✈❡• ♦❢ ❣❡♥❡r❛•✐t②✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ❢♦r ❛♥② ✜♥✐t❡ s❡q✉❡♥❝❡ (x i ) i≥0 ✐♥ X✱ ❛♥❞ ❛♥② p ∈ (1, ∞)✱ i≥0 x i ξ i Lp((0,1),X) ≈ cp i≥0 f i ξ i L2((0,1),X) ❍♦✇❡✈❡r✱ ✇❡ ❛r❡ ♠♦r❡ ✐♥t❡r❡st❡❞✱ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ✐♥t❡❣r❛✲ t✐♦♥✱ ✐♥ ✜♥❞✐♥❣ ❛♥ ❛♥❛•♦❣ ♦❢ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ♦❢ ✭✸✮✱ ✇❤✐❝❤ ✐s ♠♦r❡ ❡❛s✐•② ❝♦♠♣✉t❡❞ ♦r ❡st✐♠❛t❡❞✳ ◆♦t❡ r✐❣❤t ❛✇❛② t❤❛t t❤❡ s❛♠❡ ❡①♣r❡ss✐♦♥ ❝❛♥♥♦t ✇♦r❦ ✐♥ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❝♦♥t❡①t s✐♥❝❡ t❤❡r❡ ❛r❡ t✇♦ ♣♦ss✐❜•❡ ✧sq✉❛r❡ ❢✉♥❝t✐♦♥s✧ ✭r♦✇ ❛♥❞ ❝♦•✉♠♥✮ ✇❤✐❝❤ ❛r❡ ♥♦t ❡q✉✐✈❛•❡♥t ✐♥ L p ❢♦r p = 2✳ ▲❡t M ❜❡ ❛ ♥♦♥❝♦♠✲ ♠✉t❛t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡✱ M c t❤❡ ✜♥✐t❡•② s✉♣♣♦rt❡❞ ♦♣❡r❛t♦rs ✐♥ M ❛♥❞ S(M) t❤❡ s❡t ♦❢ ✜♥✐t❡ s❡q✉❡♥❝❡s ♦❢ M c ✳ |❡❝❛•• t❤❛t ❛ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡ E ✐s ❛ ✭q✉❛s✐✲✮❇❛♥❛❝❤ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s s✉❝❤ t❤❛t t❤❡ ♥♦r♠ ♦♥•② ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s✳ ❚❤❡s❡ s♣❛❝❡s ❝❛♥ ❜❡ tr❛♥s❢❡r❡❞ t♦ t❤❡ ♥♦♥❝♦♠♠✉✲ t❛t✐✈❡ s❡tt✐♥❣ ✇✐t❤ t❤❡ s❛♠❡ ❞❡✜♥✐t✐♦♥ ❬✸✶(✳ ❚❤✐s ❝♦♥str✉❝t✐♦♥ ❣❡♥❡r❛•✐③❡s t❤❡ ♦♥❡ ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ L p ✲s♣❛❝❡s✳ ❉❡✜♥✐t✐♦♥ ✵✳✷✳✽ ✭|♦✇ ❛♥❞ ❝♦•✉♠♥ ♥♦r♠s✮✳ ▲❡t x ∈ S(M)✱ E ❛ s②♠♠❡tr✐❝ s♣❛❝❡✳ ❉❡✜♥❡ t❤❡ ❝♦•✉♠♥ ❛♥❞ r♦✇ ♥♦r♠s ❛s ❢♦••♦✇s✿ x C E := i∈N x * i x i 1/2 E ❛♥❞ x R E := i∈N x i x * i 1/2 E ✳ ❚❤❡♦r❡♠ ✵✳✷✳✾ ✭◆♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s✮✳ ▲❡t p ∈ [2, ∞) t❤❡♥✿ i∈N x i ⊗ ξ i p ≈ max x Rp , x Cp . ▲❡t p ∈ (0, 2] t❤❡♥✿ i∈N x i ⊗ ξ i p ≈ inf y+z=x z Cp + y Rp . ❚❤✐s r❡s✉•t ✐s ❞✉❡ t♦ ▲✉st✲P✐q✉❛r❞ ❢♦r p ∈ (1, ∞) ❬✸✻(✱ ▲✉st✲P✐q✉❛r❞ ❛♥❞ P✐s✐❡r ❢♦r p = 1 ❬✸✼( ❛♥❞ ♠♦r❡ t❤❛♥ t✇❡♥t② ②❡❛rs •❛t❡r P✐s✐❡r ❛♥❞ |✐❝❛r❞ ❬✹✼( ❢♦r p < 1✳ ❇② t❤❡ r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ ♠❡t❤♦❞✱ t❤❡ r❡s✉•t ❡①t❡♥❞s t♦ ♠♦st ❇❛♥❛❝❤ ▲♦r❡♥t③ s♣❛❝❡s ❜✉t ♥♦t t♦ L 2,∞ ♦r L 1,∞ ✳ ❋♦r L 2,∞ ✱ ✐t ✐s ❞✉❡ t♦ t❤❡ ❢❛❝t t❤❛t t♦ ✇r✐t❡ L 2,∞ ❛s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ♦❢ L p ❛♥❞ L q ✱ ♦♥❡ ♠✉st ❤❛✈❡ p < 2 ❛♥❞ q > 2 ❛♥❞ t❤❡r❡❢♦r❡ ✉s❡ t❤❡ t✇♦ ❢♦r♠✉•❛s ✇❤✐❝❤ ❞♦ ♥♦t ✐♥t❡r♣♦•❛t❡ ♥✐❝❡•②✳ ❙✐♠✐•❛r•②✱ t♦ ✇r✐t❡ L 1,∞ ❛s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ♦❢ L p ❛♥❞ L q ✱ ♦♥❡ ♠✉st ❤❛✈❡ p < 1 ✇❤✐❝❤ ✐s ♣r♦❜•❡♠❛t✐❝ s✐♥❝❡ t❤❡ ❝•♦s❡❞ ✈❡❝t♦r s♣❛❝❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ξ i ✐s ♥♦t ❝♦♠♣•❡♠❡♥t❡❞ ✐♥ L p (0, 1) ❢♦r p < 1✳ ❚❤❡s❡ t✇♦ ❡①❛♠♣•❡s ✇❤✐❝❤ ❛r❡ r❡•❛t✐✈❡•② ❝♦♠♠♦♥ s♣❛❝❡s ♣r♦♠♣t❡❞ ♦✉r ✐♥✈❡st✐❣❛t✐♦♥✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t t❤❡ ξ i ❛r❡ ❛♥ ❛r❜✐tr❛r② ♦rt❤♦♥♦r♠❛• s❡q✉❡♥❝❡ ♦❢ ❡•❡♠❡♥ts ✐♥ ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♣r♦❜❛❜✐•✐t② s♣❛❝❡✳ ❲❡ ♦❜t❛✐♥ t❤❡ ❢♦••♦✇✐♥❣ t❤❡♦r❡♠✿ ❚❤❡♦r❡♠ ✵✳✷✳✶✵✳ ▲❡t p < 1✳ ❙✉♣♣♦s❡ t❤❛t ❢♦r ❛•• x ∈ S(M)✱ i≥0 x i ⊗ ξ i p inf y+z=x z Cp + y Rp . ❚❤❡♥ ❢♦r ❛♥② ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ E ❜❡t✇❡❡♥ L p ❛♥❞ L ∞ ❛♥❞ x ∈ S(M)✱ i≥ x i ⊗ ξ i E inf y+z=x z C E + y R E . ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦♥ t❤❡ •❡❢t ❤❛♥❞ s✐❞❡ ❝❛♥ ❜❡ ❝❤♦s❡♥ t♦ ❜❡ ✐♥❞❡✲ ♣❡♥❞❡♥t ♦❢ E✳ ❚❤✐s t❤❡♦r❡♠ ✐♠♣•✐❡s t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ L 1,∞ ❢♦r |❛❞❡♠❛❝❤❡r ✈❛r✐❛❜•❡s ❛s ❛ ❝♦r♦••❛r②✱ s✐♥❝❡ ❢♦r t❤♦s❡ ✈❛r✐❛❜•❡s✱ ❛s ♠❡♥t✐♦♥❡❞ ❜❡❢♦r❡✱ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ❛r❡ ✈❛•✐❞ ❢♦r p < 1✳ ❈♦r♦••❛r② ✵✳✷✳✶✶✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ξ i ❛r❡ |❛❞❡♠❛❝❤❡r ✈❛r✐❛❜•❡s ✭♦r ❢r❡❡ ❍❛❛r ✉♥✐t❛r✐❡s✮✳ ❚❤❡♥ ❢♦r ❛•• x ∈ S(M)✿ i≥0 x i ⊗ ξ i 1,∞ ≈ inf y+z=x z C1,∞ + y R1,∞ . ❚❤❡ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠ ✐s t♦ t❛❦❡ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ x = y + z ✇❤✐❝❤ ✐s ❝•♦s❡ t♦ ❜❡✐♥❣ ♦♣t✐♠❛• ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ❛•♠♦st ♠✐♥✐♠✐③❡s t❤❡ q✉❛♥✲ t✐t② y p Rp + z p Cp ❛♥❞ t♦ s❤♦✇ ❜② ❛♣♣•②✐♥❣ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ L p t♦ s❡q✉❡♥❝❡s ♦❢ t❤❡ ❢♦r♠ (ex i ) i∈N ✱ ✇❤❡r❡ e ✐s ❛ ♣r♦•❡❝t✐♦♥✱ t❤❛t t❤❡ r♦✇ K✲❢✉♥❝t✐♦♥❛• ♦❢ y ❢♦r t❤❡ ❝♦✉♣•❡ (L p , L ∞ ) ❛♥❞ t❤❡ ❝♦•✉♠♥ K✲ ❢✉♥❝t✐♦♥❛• ♦❢ z ❢♦r t❤❡ ❝♦✉♣•❡ (L p , L ∞ ) ❛r❡ ❞♦♠✐♥❛t❡❞ ❜② t❤❡ K✲❢✉♥❝t✐♦♥❛• ♦❢ i∈N x i ⊗ ξ i ❢♦r t❤❡ ❝♦✉♣•❡ (L p , L ∞ )✳ ❚❤❡♥✱ ❜② ❛♣♣•②✐♥❣ t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ▲♦r❡♥t③✲❙❤✐♠♦❣❛❦✐ t❤❡♦r❡♠✱ ✇❡ ♦❜t❛✐♥ ❛ ❝♦♥tr♦• ✐♥ ❡✈❡r② ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡✳ ❇② ♣✉s❤✐♥❣ ❢✉rt❤❡r t❤❡ ✐❞❡❛ ♦❢ st✉❞②✐♥❣ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ t❤❡ ♣❛r✲ t✐❝✉•❛r ❝❛s❡ ♦❢ L 1 ✱ ✇❡ ♦❜t❛✐♥ ❛♥ ❛•❣❡❜r❛✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ❢♦r s❡q✉❡♥❝❡s ✇❤✐❝❤ s❡❡♠s t♦ ♣•❛② t❤❡ r♦•❡ ♦❢ ❛ ♣♦•❛r ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❡♥❛❜•❡s ✉s t♦ r❡❝♦✈❡r ♠♦st ❦♥♦✇♥ r❡s✉•ts ♦♥ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❢r❡❡ ❍❛❛r ✉♥✐t❛r✐❡s✳ ❚❤❡♦r❡♠ ✵✳✷✳✶✷✳ ▲❡t x = (x i ) i≥0 ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ❡•❡♠❡♥ts ✐♥ M c ✳ ❚❤❡r❡ ❡①✐st α, β ∈ M + c ✱ ❛♥❞ u = (u i ) i≥0 ∈ M N c s✉❝❤ t❤❛t✿ • s(α) ≤ i≥0 u i u * i ≤ 1✱ • s(β) ≤ i≥0 u * i u i ≤ 1✱ • ❢♦r ❛•• i ≥ 0✱ x i = u i β + αu i ✳ ❋✉rt❤❡r♠♦r❡✱ s✉♣♣♦s❡ t❤❛t✿ • t❤❡ ξ i ❛r❡ ♦rt❤♦♥♦r♠❛• ✐♥ L 2 (A) • t❤❡ ξ i ✈❡r✐❢② t❤❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t② ✐♥ L ∞ ✐✳❡ ❢♦r ❛♥② ✜♥✐t❡ s❡q✉❡♥❝❡ x = (x i ) i≥0 ∈ M N ✿ max x R∞ , x C∞ ≈ c∞ i≥0 x i ⊗ ξ i ∞ . ❚❤❡♥ ❢♦r ❛♥② p ∈ (0, ∞) ❛♥❞ E ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ L p ❛♥❞ L ∞ ✿ α E + β E ≈ c i≥0 x i ⊗ ξ i E , ✇❤❡r❡ c ♦♥•② ❞❡♣❡♥❞s ♦♥ p ❛♥❞ c ∞ ✳ ✵✳✷✳ ■◆❚|❖❉❯❈❚■❖◆ ✭❊◆•▲■❙❍✮ ✷✸ ❚❤✐s t❤❡♦r❡♠ ✐s ♣r♦✈❡❞ ✐♥ t✇♦ ✐♥❞❡♣❡♥❞❡♥t st❡♣s✳ ❋✐rst✱ t❤❡ ❛•❣❡❜r❛✐❝ ❞❡✲ ❝♦♠♣♦s✐t✐♦♥ ✐s ❢♦✉♥❞ ❜② ❝♦♥s✐❞❡r✐♥❣ ❛♥ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ x = y + z ✐♥ L 1 ✳ ❇② ✇r✐t✐♥❣ t❤❡ ♣♦•❛r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ y ❛♥❞ z ❛♥❞ t❤❡✐r ♦♣t✐♠❛•✐t② ✐♥ t❡r♠s ♦❢ ❞✉❛•✐t②✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❛•❣❡❜r❛✐❝ ✐❞❡♥t✐t✐❡s ✇❡ ♥❡❡❞✳ ❚❤❡♥✱ ✉s✐♥❣ t❤❡s❡ ❛•❣❡✲ ❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥✱ ✇❡ ♣r♦✈❡ ♦♥❝❡ ❛❣❛✐♥ ❛ ❝♦♥tr♦• ✐♥ t❡r♠s ♦❢ K✲❢✉♥❝t✐♦♥❛•s ✇❤✐❝❤ ❞❡•✐✈❡rs t❤❡ r❡s✉•t ❝•❛✐♠❡❞ ✐♥ t❤❡ t❤❡♦r❡♠✳ ❋♦r p ≥ 1✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ❞♦ ♥♦t r❡q✉✐r❡ ❛♥② ❝♦♠♣•❡① t♦♦• ❜✉t t♦ tr❡❛t t❤❡ ❝❛s❡ ♦❢ p < 1✱ ✇❡ ❝r✉❝✐❛••② ♥❡❡❞ ❛ ❢♦r♠ ♦❢ ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t② ❢♦r ❝♦♠♠✉t❛t♦rs r❡❝❡♥t•② ♣r♦✈❡❞ ❜② |✐❝❛r❞ ❬✺✹(✳ ◆♦t❡ t❤❛t t❤❡ t❤❡♦r❡♠ ❣✐✈❡s ❛ ❞❡t❡r♠✐♥✐st✐❝ ❡q✉✐✈❛•❡♥t ♦❢ t❤❡ ♥♦r♠ ♦❢ i≥0 x i ⊗ ξ i ✐♥ L 2,∞ t❤♦✉❣❤ ✐ts ✈❡r② ♣❛rt✐❝✉•❛r ❢♦r♠ ♠❛② ♥♦t ♠❛❦❡ ✐t s✉✐t❛❜•❡ ❢♦r ❛♣♣•✐❝❛t✐♦♥s✳ ❆♥♦t❤❡r ♥♦t❛❜•❡ ❢❛❝t ✐s t❤❛t t❤❡ ❢♦r♠✉•❛ ❢♦r t❤✐s ❞❡t❡r♠✐♥✐st✐❝ ❡q✉✐✈❛•❡♥t✱ ❝♦♥tr❛r② t♦ ♣r❡✈✐♦✉s•② ❦♥♦✇♥ ♦♥❡s✱ ✉♥✐✜❡s t❤❡ ❝❛s❡s p < 2 ❛♥❞ p > 2✳ ❋✐♥❛••②✱ ✇❡ ❣✐✈❡✱ ✉♥❞❡r s♦♠❡ t❡❝❤♥✐❝❛• ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ s♣❛❝❡s ❛♥❞ ✐♥ t❤❡ s❡tt✐♥❣ ♦❢ ❢r❡❡ ❍❛❛r ✉♥✐t❛r✐❡s✱ ❛ ❝❤❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ s♣❛❝❡s ✐♥ ✇❤✐❝❤ t❤❡ ✉s✉❛• ❢♦r♠✉•❛s ❢♦r ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ❤♦•❞✳ ❚❤❡♦r❡♠ ✵✳✷✳✶✸✳ ▲❡t E ❜❡ ❛ s②♠♠❡tr✐❝ s♣❛❝❡ ✇✐t❤ t❤❡ ❋❛t♦✉ ♣r♦♣❡rt② ❛♥❞ M = B(ℓ 2 )⊗L ∞ (0, 1)✳ ❚❤❡♥ E ✐s •❡❢t✲2✲♠♦♥♦t♦♥❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ ❢♦r ❛•• ✜♥✐t❡ s❡q✉❡♥❝❡s x ∈ M N c ✱ x R E ∩C E ≈ i≥0 x i ⊗ ξ i E ✳ ❙✉♣♣♦s❡ ❢✉rt❤❡r♠♦r❡ t❤❡r❡ ❡①✐sts p > 0 s✉❝❤ t❤❛t E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ L p ❛♥❞ L ∞ ✳ ❚❤❡♥ E ✐s r✐❣❤t✲2✲♠♦♥♦t♦♥❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ ❢♦r ❛•• ✜♥✐t❡ s❡q✉❡♥❝❡s x ∈ M N c ✱ x R E +C E ≈ i≥0 x i ⊗ ξ i E ✳ ❚❤✐s ♣r♦✈✐❞❡s ❛ ❝♦♠♣•❡♠❡♥t t♦ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❢♦r♠✉•❛ ❣✐✈❡♥ ❜② t❤❡ ♣r❡✲ ✈✐♦✉s t❤❡♦r❡♠✳ ❙❤♦✇✐♥❣ ❢♦r ❡①❛♠♣•❡ t❤❛t ✐♥ t❤❡ ❝❛s❡ ♦❢ E = L 2,∞ ✱ ♥♦♥❡ ♦❢ t❤❡ ✉s✉❛• ❢♦r♠✉•❛s ❝❛♥ ✇♦r❦✳ ❲❡❛❦ t②♣❡ ✐♥❡q✉❛•✐t② ❢♦r s✐♥❣✉•❛r ✐♥t❡❣r❛•s ■♥ t❤❡ •❛st ❝❤❛♣t❡r✱ ✇❡ ♣r❡s❡♥t t❤❡ ✇♦r❦ ✇❤✐❝❤ ❛♣♣❡❛rs ✐♥ ❬✺(✳ ■t ✐s ❛•♠♦st ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ t❤❡ r❡st ♦❢ t❤❡ t❡①t✱ ❛♣❛rt ❢r♦♠ ❛ s❤♦rt ❛♣♣•✐❝❛t✐♦♥ ♦❢ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ❛♣♣❡❛r✐♥❣ ❛t t❤❡ ❡♥❞✳ ■t ❞❡❛•s ✇✐t❤ s✐♥❣✉•❛r ✐♥t❡❣r❛•s ✐✳❡✳ ❛ ❝❡rt❛✐♥ t②♣❡ ♦❢ ♦♣❡r❛t♦r T ✐♥❢♦r♠❛••② ❞❡✜♥❡❞ ❜② ❛♥ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❢♦r♠✿ T f (x) = R n k(x, y)f (y)dy, ✇❤❡r❡ n ≥ 1✱ x ∈ R n ✱ k : R 2n → C ✐s s❛✐❞ t♦ ❜❡ t❤❡ ❦❡r♥❡• ♦❢ T ❛♥❞ ✈❡r✐✜❡s ❝♦♥❞✐t✐♦♥s t❤❛t ❛••♦✇ ❛ s✐♥❣✉•❛r✐t② ♦♥ t❤❡ ❞✐❛❣♦♥❛• ✭♥❡❛r x = y✮ ❜✉t ❝♦♥tr♦• ✐ts s♣❡❡❞ ♦❢ ✈❛r✐❛t✐♦♥ ✭✇❤✐❝❤ ✇❡ ✇✐•• ❝❛•• ❈❛•❞❡ró♥✲)②❣♠✉♥❞ t②♣❡ ❝♦♥❞✐t✐♦♥s✮✳ ■❢ t❤❡r❡ ❡①✐sts ❛ ❢✉♥❝t✐♦♥ K : R n → C s✉❝❤ t❤❛t k(x, y) = K(x -y)✱ k ✐s s❛✐❞ t♦ ❜❡ ♦❢ ❝♦♥✈♦•✉t✐♦♥ t②♣❡ ❛♥❞ ✐♥ t❤✐s ❝❛s❡✱ ✐❢ K ∈ C 1 (R n \ {0})✱ ❛ t②♣✐❝❛• ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ❝♦♥❞✐t✐♦♥ ✇♦✉•❞ ❜❡✿ |∇K(x)| 1 |x| n+1 . ❚❤✐s ❝♦♥❞✐t✐♦♥ ✐s ✈❡r✐✜❡❞ ❜② t❤❡ |✐❡s③ ❛♥❞ t❤❡ ❍✐•❜❡rt tr❛♥s❢♦r♠ ✇❤✐❝❤ ❛r❡ s♦♠❡ ♦❢ t❤❡ ♠♦t✐✈❛t✐♥❣ ❡①❛♠♣•❡s ❢♦r ✐♥tr♦❞✉❝✐♥❣ t❤❡ ♥♦t✐♦♥ ♦❢ s✐♥❣✉•❛r ✐♥t❡❣r❛•✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❣❡♥❡r❛•✐s❛t✐♦♥s ♦❢ t❤❡ ❢♦••♦✇✐♥❣ t❤❡♦r❡♠✱ t❤❛t ✇❡ st❛t❡ ❤❡r❡ ✐♥❢♦r♠❛••②✳ ❚❤❡♦r❡♠ ✵✳✷✳✶✹ ✭❈❛•❞❡ró♥✱ )②❣♠✉♥❞✮✳ ■❢ T ✐s ❜♦✉♥❞❡❞ ♦♥ L 2 (R n ) ❛♥❞ ✐s ❛ss♦❝✐❛t❡❞ t♦ ❛ ❦❡r♥❡• k ✈❡r✐❢②✐♥❣ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ t②♣❡ ❝♦♥❞✐t✐♦♥s t❤❡♥ T ❡①t❡♥❞s t♦ ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r ❢r♦♠ L 1 (R n ) t♦ L 1,∞ (R n )✳ ❚❤✐s t❤❡♦r❡♠ ✐♠♣•✐❡s ❜② ✐♥t❡r♣♦•❛t✐♦♥ ❛♥❞ ❞✉❛•✐t② t❤❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ s✐♥❣✉✲ •❛r ✐♥t❡❣r❛•s ❢r♦♠ L p (R n ) t♦ L p (R n ) ❢♦r ❛•• p ∈ (1, ∞)✳ ❇② t❤❡ s❛♠❡ ❡①♣r❡ss✐♦♥ T ❝♦✉•❞ ❞❡✜♥❡ ❛♥ ♦♣❡r❛t♦r ❢r♦♠ L p (R n , X) t♦ L p (R n , X) ❢♦r ❛♥② ❇❛♥❛❝❤ s♣❛❝❡ X✳ ❈♦♥❞✐t✐♦♥s ♦♥ X ❢♦r t❤❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ T t♦ r❡♠❛✐♥ tr✉❡ ✇❤❡♥ ❝♦♥s✐❞❡r✲ ✐♥❣ X✲✈❛•✉❡❞ ❢✉♥❝t✐♦♥s ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞✱ t❤❡ ❦❡② ❝♦♥❞✐t✐♦♥ ❜❡✐♥❣ t❤❛t X ✐s ❯▼❉✳ ◆♦♥❝♦♠♠✉t❛t✐✈❡ L p ✲s♣❛❝❡s ❛r❡ ❯▼❉ ❢♦r p ∈ (1, ∞) ❛♥❞ ❤❡♥❝❡ s✐♥❣✉✲ •❛r ✐♥t❡❣r❛•s ❝❛♥ ❜❡ ✉s❡❞ ✐♥ t❤✐s ❝♦♥t❡①t✱ ❤♦✇❡✈❡r ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❛♥❛•♦❣ ♦❢ ❈❛•❞❡ró♥ ❛♥❞ )②❣♠✉♥❞ t❤❡♦r❡♠ ✇❛s ♠✐ss✐♥❣ t♦ ♦❜t❛✐♥ ♦♣t✐♠❛• ❜♦✉♥❞s ❢♦r t❤❡ ♥♦r♠ ♦❢ T ❢r♦♠ L p t♦ L p ✇❤❡♥ p t❡♥❞s t♦ 1✳ ■♥ ❬✹✶(✱ P❛r❝❡t ❣❡♥❡r❛•✐③❡❞ t❤❡♦r❡♠ ✵✳✷✳✶✹ t♦ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❝♦♥t❡①t✳ ◆♦t❡ t❤❛t ✐♥ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s❡t✲ t✐♥❣✱ ✇❤❡♥ st✉❞②✐♥❣ s✐♥❣✉•❛r ✐♥t❡❣r❛•s✱ ❇▼❖ ♠❡t❤♦❞s ♦r ❞✐r❡❝t ♣r♦♦❢s ❝❛♥ ❜❡ ♣r❡❢❡r❡❞ t♦ ✇✐t❤ ✇❡❛❦ t②♣❡ ✐♥❡q✉❛•✐t✐❡s ✇❤✐❝❤ ❝❛♥ ❜❡ t❡❝❤♥✐❝❛•✳ ◆♦♥❡t❤❡•❡ss✱ t❤❡ t❤❡♦r❡♠ ❤❛s ♥♦✇ ❢♦✉♥❞ ✈❛r✐♦✉s ❛♣♣•✐❝❛t✐♦♥s✱ ❛♠♦♥❣ ✇❤✐❝❤ ❛♥ ✐♠♣♦rt❛♥t ♦♥❡ ✐♥ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r②✳ ❚❤❡ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ❛♥❞ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❤❛r♠♦♥✐❝ ❛♥❛•②s✐s st❛rt❡❞ ✐♥ ❬✺✶( ✇❤❡r❡ P♦t❛♣♦✈ ❛♥❞ ❙✉❦♦❝❤❡✈ s♦•✈❡❞ ❛ •♦♥❣st❛♥❞✐♥❣ ♣r♦❜✲ •❡♠ ♦♥ ♦♣❡r❛t♦r ▲✐♣s❤✐t③ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ ❙❝❤❛tt❡♥ ❝•❛ss❡s✳ ❚❤❡② ♣r♦✈❡❞ t❤❛t ❢♦r ❛♥② p ∈ (1, ∞)✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c p s✉❝❤ t❤❛t ❢♦r ❛♥② ▲✐♣s❝❤✐t③ ❢✉♥❝t✐♦♥ f : R → R ❛♥❞ s❡•❢✲❛❞•♦✐♥t ♦♣❡r❛t♦rs x ❛♥❞ y✱ f (x) -f (y) p ≤ c p f Lip x -y p , ✭✹✮ ✇❤❡r❡ ✇❡ ❛ss✉♠❡ t❤❛t x -y ∈ L p (M) ❛♥❞✿ f Lip := sup t =s |f (t) -f (s)| |t -s| . |❡❝❡♥t•②✱ t❤❡ t✇♦ ❛✉t❤♦rs✱ •♦✐♥❡❞ ❜② ❈❛s♣❡rs ❛♥❞ )❛♥✐♥✱ ❣❛✈❡ ❛ ♥❡✇ ♣r♦♦❢ ♦❢ ✭✹✮ ✇❤✐❝❤ ❞❡s❝r✐❜❡s t❤❡ ♦♣t✐♠❛• ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ❝♦♥st❛♥t c p ✇❤❡♥ p t❡♥❞s t♦ 1✳ ■t r❡•✐❡s ♦♥ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ t❤❡♦r② ❛♥❞ ✐♥ ♣❛rt✐❝✉•❛r ♦♥ P❛r❝❡t✬s t❤❡♦r❡♠✳ ❚❤❡ s❝❤❡♠❡ ♦❢ ♣r♦♦❢ t❤❡② ❡♠♣•♦② ✲ t♦ r❡❞✉❝❡ ❛ q✉❡st✐♦♥ t♦ t❤❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r ✲ ❝♦✉•❞ ❜❡ ❛♣♣•✐❝❛❜•❡ t♦ ♦t❤❡r ♣r♦❜•❡♠s✳ ❚❤✐s ✐s t❤❡ ♠❛✐♥ r❡❛s♦♥ ✇❤② ❛ ♣❛rt✐❝✉•❛r ✐♥t❡r❡st ❤❛s s♣r✉♥❣ t♦✇❛r❞s P❛r❝❡t✬s t❤❡♦r❡♠ ✐♥ t❤❡ •❛st ❢❡✇ ②❡❛rs✳ ❍♦✇❡✈❡r✱ t❤❡ ♣r♦♦❢ ❣✐✈❡♥ ✐♥ ❬✹✶( ✐s ❞✐✣❝✉•t ❛♥❞ t❤❡r❡❢♦r❡ ♥♦t ❡❛s② ❛❞❛♣t ✐❢ ♦♥❡ s❤♦✉•❞ ❡♥❝♦✉♥t❡r ❛ s✐t✉❛t✐♦♥ ✇❤❡r❡ t❤❡ t❤❡♦r❡♠ ❞♦❡s ♥♦t ✐♠♠❡❞✐❛t❡•② ❛♣♣•②✳ ❚❤✐s ✐s ✇❤❛t ♠♦t✐✈❛t❡❞ ♦✉r ✇♦r❦✳ ❲❡ ♣r❡s❡♥t ❛ ♥❡✇ s✐♠♣•✐✜❡❞ ♣r♦♦❢ ♦❢ t❤❡ ✇❡❛❦ ❜♦✉♥❞❡❞♥❡ss ♦❢ s✐♥❣✉•❛r ✐♥t❡❣r❛•s✱ r❡•②✐♥❣ ♦♥ t❤❡ s❛♠❡ ✐❞❡❛ ❛s P❛r❝❡t✬s✳ ▲❡t ✉s ✜rst ❣✐✈❡ ❛ ♣r❡❝✐s❡ st❛t❡♠❡♥t ♦❢ t❤❡ t❤❡♦r❡♠✳ ❲❡ st❛rt ❜② ✐♥tr♦❞✉❝✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s ✇❡ ✐♠♣♦s❡ ♦♥ t❤❡ ❦❡r♥❡•✳ ❉❡✜♥✐t✐♦♥ ✵✳✷✳✶✺ ✭❙♠♦♦t❤♥❡ss ❛♥❞ ❙✐③❡✮✳ ❙❛② t❤❛t T ❤❛s ▲✐♣s❝❤✐t③ ♣❛r❛♠❡t❡r γ ✭0 < γ ≤ 1✮ ✐❢ ❢♦r ❛•• x✱y ❛♥❞ z ✐♥ R n ✈❡r✐❢②✐♥❣ |x -z| ≤ 1 2 |y -z|✱ t❤❡ ❢♦••♦✇✐♥❣ s♠♦♦t❤♥❡ss ❡st✐♠❛t❡s ❤♦•❞✿ |k(x, y) -k(z, y)| |x -z| γ |y -z| n+γ , |k(y, x) -k(y, z)| |x -z| γ |y -z| n+γ .
❲❡ s❛② t❤❛t T ✈❡r✐✜❡s t❤❡ s✐③❡ ❡st✐♠❛t❡ ✐❢ ❢♦r ❛

•• x, y ∈ R n ✱ k(x, y) M 1 |x -y| n . ❚❤❡♦r❡♠ ✵✳✷✳✶✻✳ ❆ss✉♠✐♥❣ t❤❛t T ❤❛s ❛ ▲✐♣s❝❤✐t③ ♣❛r❛♠❡t❡r γ ✭0 < γ ≤ 1✮✱ ✈❡r✐✜❡s t❤❡ s✐③❡ ❡st✐♠❛t❡ ❛♥❞ ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L 2 (R n ) t♦ L 2 (R n )✳ ❚❤❡♥ T ❡①t❡♥❞s ❛s ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r ❢r♦♠ L 1 (M⊗L ∞ (R n )) t♦ L 1,∞ (M⊗L ∞ (R n ))✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤✐s t❤❡♦r❡♠✱ ✐♥ t❤❡ ❝•❛ss✐❝❛• s❡tt✐♥❣✱ r❡•✐❡s ♦♥ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s f = g + b✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ✐♥ t❤❡ •❛♥❣✉❛❣❡ ♦❢ ♠❛rt✐♥✲ ❣❛•❡s✳ ❚❤✐s ✐♥t❡r♣r❡t❛t✐♦♥ ❛••♦✇s t♦ ❝♦♥str✉❝t ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ✈❡rs✐♦♥ ♦❢ t❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥✱ ❜② ✉s✐♥❣ t❤❡ ✇❡••✲❞❡✈❡•♦♣❡❞ t❤❡♦r② ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❛rt✐♥✲ ❣❛•❡s ❛♥❞ ✐♥ ♣❛rt✐❝✉•❛r ❈✉❝✉•❡s❝✉✬s ♣r♦•❡❝t✐♦♥s✳ ❚❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s❡tt✐♥❣ ❤♦✇❡✈❡r✱ ❞♦❡s ♥♦t ❡♥•♦② ❛•• t❤❡ ❣♦♦❞ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❝•❛ss✐❝❛• ♦♥❡✳ ◆❡✇ t❡r♠s ❛♣♣❡❛r ❛♥❞ t♦ ❡st✐♠❛t❡ t❤❡♠✱ t❤❡ ❢♦••♦✇✐♥❣ L 2 ✲♣s❡✉❞♦•♦❝❛•✐s❛t✐♦♥ t❤❡♦r❡♠ ✐s ♥❡❡❞❡❞✳ ❉❡♥♦t❡ ❜② N t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ M⊗L ∞ (R n ) ❛♥❞ ❜② N k t❤❡ s✉❜❛•❣❡❜r❛ ♦❢ N ❝♦♠♣♦s❡❞ ♦❢ ❢✉♥❝✲ t✐♦♥s ❝♦♥st❛♥t ♦♥ ❞②❛❞✐❝ ❝✉❜❡s ♦❢ ❡❞❣❡ •❡♥❣t❤ 2 -k ✱ k ∈ Z✳ ❚❤❡♦r❡♠ ✵✳✷✳✶✼ ✭L 2 ✲♣s❡✉❞♦•♦❝❛•✐s❛t✐♦♥✮✳ ▲❡t f ∈ L 2 (N ) ❛♥❞ s ∈ N✳ ❋♦r ❛•• k ∈ Z✱ •❡t A k ❛♥❞ B k ❜❡ ♣r♦•❡❝t✐♦♥s ✐♥ N k s✉❝❤ t❤❛t A ⊥ k df k+s = df k+s B ⊥ k = 0✳ ❋♦r ❛♥② ♦❞❞ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r d✱ ✇r✐t❡✿ A k = Q∈Q k A Q I Q ✱ A Q ∈ M ❛♥❞ ❞❡✜♥❡ dA k := Q∈Q k A Q I dQ ✳ ❉❡✜♥❡ dB k t❤❡ s❛♠❡ ✇❛②✳ ▲❡t✿ A f,s := k∈Z 5A k ❛♥❞ B f,s := k∈Z 5B k . ▲❡t T ❜❡ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❦❡r♥❡• k ✇✐t❤ ▲✐♣s❝❤✐t③ ♣❛r❛♠❡t❡r γ✱ ✈❡r✐❢②✐♥❣ t❤❡ s✐③❡ ❝♦♥❞✐t✐♦♥✱ t❛❦✐♥❣ ✈❛•✉❡s ✐♥ C ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ L 2 (N ) t♦ L 2 ( N )✳ ❚❤❡♥ ❢♦r ❛•• s ∈ N ❛♥❞ f ∈ L 2 (N ) ✇❡ ❤❛✈❡✿ A ⊥ f,s (T f ) 2 2 -γs 2 f 2 ❛♥❞ (T f )B ⊥ f,s 2 2 -γs 2 f 2 . ❚❤✐s t❤❡♦r❡♠ ✐s st❛t❡❞ ❤❡r❡ ❢♦r ❢✉♥❝t✐♦♥s t❛❦✐♥❣ ✈❛•✉❡s ✐♥ ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ ❜✉t ✐t ✐s ❝♦♠♠✉t❛t✐✈❡ ✐♥ ♥❛t✉r❡✳ ■t ❣✐✈❡s ❛ q✉❛♥t✐✜❡❞ ❡st✐♠❛t❡ ♦❢ t❤❡ r❛t❡ ♦❢ ❞❡❝r❡❛s❡ ♦❢ s✐♥❣✉•❛r ✐♥t❡❣r❛•s ♦✉ts✐❞❡ ♦❢ t❤❡ ❞✐❛❣♦♥❛•✳ ❚❤✐s ❤❛❞ ♥❡✈❡r ❜❡❡♥ ♥❡❡❞❡❞ ❜❡❢♦r❡ ✐♥ t❤❡ ❝•❛ss✐❝❛• t❤❡♦r② ❜✉t s✉❝❤ ❡st✐♠❛t❡s ❛r❡ ❝r✉❝✐❛• ✇❤❡♥ ❞❡❛•✐♥❣ ✇✐t❤ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❝❛s❡✳ ❚❤❡ ♣r♦♦❢ ♦❢ L 2 ✲♣s❡✉❞♦•♦❝❛•✐s❛t✐♦♥ ❜♦✐•s ❞♦✇♥ t♦ s❤♦✇✐♥❣ ❛♥ ❛•♠♦st ♦rt❤♦❣♦♥❛•✐t② r❡s✉•t✳ ■♥ t❤✐s t❡①t✱ ♦♥•② ❡•❡♠❡♥t❛r② t❡❝❤♥✐q✉❡s ❛r❡ ❡♠♣•♦②❡❞✿ ❙❝❤✉r✬s •❡♠♠❛ t♦ ❡st✐♠❛t❡ t❤❡ L 2 → L 2 ♥♦r♠ ♦❢ ❛ ❦❡r♥❡• ♦♣❡r❛t♦r ❛♥❞ ❛ ❝❛r❡❢✉• ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ T ✳ ❚❤❡ ♠❛✐♥ ♥♦✈❡•t② ♦❢ ♦✉r ❛♣♣r♦❛❝❤ ❝♦♠♣❛r❡❞ t♦ P❛r❝❡t✬s ✐s t❤❛t ✇❡ ✐♥t❡r♣r❡t t❤❡ ❝♦♠♣✉t❛t✐♦♥s ♠❛❞❡ ✐♥ t❡r♠s ♦❢ ✧❦❡r♥❡• ♠♦❞✐✜❝❛t✐♦♥s✧ ✐✳❡✳ ✐❢ ✇❡ ♠❛♥❛❣❡ t♦ ♣r♦✈❡ t❤❛t A f,s (x)T f (x) = A f,s (x) R n k(x, y)f (y)dy = A f,s (x) R n k ′ (x, y)f (y)dy, ✇❡ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ ♦♣❡r❛t♦r T ′ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❦❡r♥❡• k ′ ✳ ❚❤✐s ❛••♦✇s t♦ ✇♦r❦ ❜♦t❤ ♦♥ t❤❡ •❡✈❡• ♦❢ ♦♣❡r❛t♦rs ❛♥❞ ❦❡r♥❡•s t❤✉s t♦ ✐♠♣r♦✈❡ t❤❡ r❡❛❞❛❜✐•✐t② ♦❢ t❤❡ ♣r♦♦❢ ❛♥❞ t♦ s❤♦rt❡♥ ✐t✳ ❆s ❛ ❜②♣r♦❞✉❝t ♦❢ t❤✐s s✐♠♣•✐✜❝❛t✐♦♥✱ ✇❡ ❝❛♥ q✉✐❝❦•② r❡❝♦✈❡r ❛ ✈❡rs✐♦♥ ♦❢ L p ✲♣s❡✉❞♦•♦❝❛•✐s❛t✐♦♥✱ ❛ r❡s✉•t s✐♠✐•❛r t♦ t❤❡ ♦♥❡ ♦❜t❛✐♥❡❞ ❜② ❍②tö♥❡♥ ✐♥ ❬✷✷(✳ ❚❤❡♦r❡♠ ✵✳✷✳✶✽✳ ❯s✐♥❣ t❤❡ ♥♦t❛t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❛❜♦✈❡✱ •❡t T ❜❡ ❛ ❈❛•❞❡ró♥✲ )②❣♠✉♥❞ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❦❡r♥❡• k ✇✐t❤ ▲✐♣s❝❤✐t③ ♣❛r❛♠❡t❡r γ✱ ✈❡r✐✲ ❢②✐♥❣ t❤❡ s✐③❡ ❝♦♥❞✐t✐♦♥✱ t❛❦✐♥❣ ✈❛•✉❡s ✐♥ C ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ L 2 (N ) t♦ L 2 ( N )✳ ❚❤❡♥ ❢♦r ❛•• 1 < p < ∞✱ s ∈ N ❛♥❞ f ∈ L p (N )✱ t❤❡r❡ ❡①✐sts θ p > 0 s✉❝❤ t❤❛t ✿ A ⊥ f,s (T f ) p 2 -γsθp f p ❛♥❞ (T f )B ⊥ f,s p 2 -γsθp f p , ✇❤❡r❡ t❤❡ ✐♠♣•✐❡❞ ❝♦♥st❛♥t ❞❡♣❡♥❞s ♦♥ p✳ ❚❤❡ s✐♠♣•✐✜❝❛t✐♦♥s ♠❛❞❡ ❛•s♦ ❛••♦✇ ✉s t♦ ♣r❡❝✐s❡•② tr❛❝❦ t❤❡ ✉s❡ ♦❢ ❡❛❝❤ ❤②♣♦t❤❡s✐s ❞✉r✐♥❣ t❤❡ ♣r♦♦❢✳ ❲❡ ♦❜t❛✐♥ t❤✐s ✇❛② ❛ ✈❡rs✐♦♥ ♦❢ t❤❡ t❤❡♦r❡♠ ❢♦r ♦♣❡r❛t♦r✲✈❛•✉❡❞ ❦❡r♥❡•s✱ ❛s •♦♥❣ ❛s t❤❡ ✈❛•✉❡s t❛❦❡♥ ❜② t❤❡ ❦❡r♥❡• ❝♦♠♠✉t❡ ✇✐t❤ t❤❡ ✈❛•✉❡s t❛❦❡♥ ❜② t❤❡ ❢✉♥❝t✐♦♥✳ ■t ✐s ❜✉t ❛ s♠❛•• ✐♠♣r♦✈❡♠❡♥t ♦❢ t❤❡ ♦r✐❣✐♥❛• ♣❛♣❡r ❜✉t ❝♦♠❜✐♥✐♥❣ ✐t ✇✐t❤ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ L 1,∞ ✱ ✇❡ r❡❝♦✈❡r ✇✐t❤♦✉t ❡✛♦rt t❤❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ s✐♥❣✉•❛r ✐♥t❡❣r❛•s ✇✐t❤ ❛ ❍✐•❜❡rt✲✈❛•✉❡❞ ❦❡r♥❡•✱ ♦r✐❣✐♥❛••② ♦❜t❛✐♥❡❞ ❜② ▼❡✐ ❛♥❞ P❛r❝❡t ✐♥ ❬✸✾(✳ ❚❤❡♦r❡♠ ✵✳✷✳✶✾✳ ▲❡t T ❜❡ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ t♦ ❛ ❦❡r♥❡• k t❛❦✐♥❣ ✈❛•✉❡s ✐♥ ℓ 2 ✳ ❙✉♣♣♦s❡ ❢✉rt❤❡r♠♦r❡ t❤❛t T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L 2 (N ) t♦ R 2 (N ) ❛♥❞ t❤❛t k s❛t✐s✜❡s t❤❡ s♠♦♦t❤♥❡ss ❛♥❞ s✐③❡ ❝♦♥❞✐t✐♦♥✳ ❚❤❡♥ T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L 1 (N ) t♦ R 1,∞ (N ) + C 1,∞ (N ) ❛♥❞ ❢r♦♠ L p (N ) t♦ R p (N ) + C p (N ) ❢♦r ❈❤❛♣t❡r ✶ ■♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ❜❡t✇❡❡♥ L p ✲s♣❛❝❡s ✶✳✶ ■♥tr♦❞✉❝t✐♦♥ ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ st✉❞② ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ❜❡t✇❡❡♥ L p ✲ s♣❛❝❡s✳ ❖♥❡ ♦❢ t❤❡ ✜rst r❡s✉•ts ♦❢ t❤✐s ❦✐♥❞ ✐s ❞✉❡ t♦ ❈❛•❞❡ró♥ ❬✻( ❛♥❞ ❞❡❛•s ✇✐t❤ t❤❡ ❝♦✉♣•❡ (L 1 , L ∞ )✳ ▲♦r❡♥t③ ❛♥❞ ❙❤✐♠♦❣❛❦✐ ❡①t❡♥❞❡❞ t❤❡ r❡s✉•t t♦ t❤❡ ❝♦✉♣•❡s (L 1 , L p ) ❛♥❞ (L p , L ∞ ) ❬✸✺(✳ ❙✐♥❝❡ t❤❡♥✱ t♦♦•s ❤❛✈❡ ❜❡❡♥ ❞❡✈❡•♦♣❡❞ t♦ ❣✐✈❡✱ ✇❤❡♥ ♣♦ss✐❜•❡✱ s②st❡♠❛t✐❝ ❞❡s❝r✐♣t✐♦♥s ♦❢ t❤❡ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ♦❢ ❛ ❣✐✈❡♥ ❝♦✉♣•❡ ♦❢ ❇❛♥❛❝❤ s♣❛❝❡s✱ s❡❡ ❢♦r ❡①❛♠♣•❡ ❬✷(✳ ■♥ t❤❡ s✐♠♣•❡st ❝❛s❡s✱ t❤❡ ❦♥♦✇•❡❞❣❡ ♦❢ t❤❡ K✲❢✉♥❝t✐♦♥❛• ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❝♦✉♣•❡ ✐s ❡♥♦✉❣❤ t♦ ♣r♦✈✐❞❡ s❛t✐s❢②✐♥❣ r❡s✉•ts✳ ❖♥❡ ♦❢ t❤❡ ♠♦t✐✈❛t✐♦♥s ❜❡❤✐♥❞ ♠❛♥② ♦❢ t❤❡ st❛t❡♠❡♥ts ❝♦♠♣✐•❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r ✐s t❤❛t ✇❡ ❡♥❝♦✉♥t❡r❡❞ s♦♠❡ ❞✐✣❝✉•t✐❡s ✐♥ ✜♥❞✐♥❣ s✉✐t❛❜•❡ r❡❢❡r❡♥❝❡s t❤❛t ❞❡❛• ✇✐t❤ t❤❡ ❝❛s❡ ♦❢ q✉❛s✐✲❇❛♥❛❝❤ s♣❛❝❡s✳ ❙♦♠❡ r❡s✉•ts ❝❛♥ ❜❡ ❣❡♥❡r❛•✐s❡❞ ✇✐t❤♦✉t ♠♦❞✐✜❝❛t✐♦♥s t♦ t❤❡ ♣r♦♦❢s✱ ♦t❤❡rs r❡q✉✐r❡ ♠♦r❡ ✇♦r❦ ❛♥❞ s♦♠❡ ❜❡❝♦♠❡ ❢❛•s❡✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✇❡ ❣❡♥❡r❛•✐s❡ t❤❡ ▲♦r❡♥t③✲❙❤✐♠♦❣❛❦✐ t❤❡♦r❡♠ t♦ t❤❡ ❝♦✉♣•❡ (L p (0, α), L ∞ (0, α))✱ p < 1✱ α ∈ (0, ∞]✳ ❆ r❡s✉•t t❤❛t ❞♦❡s ♥♦t r❡q✉✐r❡ ♥❡✇ t❡❝❤♥✐q✉❡s ❜✉t t❤❛t ✇❡ ❝♦✉•❞ ♥♦t ✜♥❞ ✐♥ t❤❡ •✐t❡r❛t✉r❡✳ ❚❤❡ s❡❝♦♥❞ ♠♦t✐✈❛t✐♦♥ ❢♦r t❤✐s ❝❤❛♣t❡r ✐s t♦ ♣r♦✈✐❞❡ ❛ ♥❡✇ ❝❤❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ❢♦r t❤❡ ❝♦✉♣•❡ (L p (0, α), L ∞ (0, α))✱ ✇❤✐❝❤ ✇✐•• ❜❡ ✉s❡❞ ✐♥ ❝❤❛♣t❡r ✸✳ ■t ✐s ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ p✲♠♦♥♦t♦♥✐❝✐t②✱ ❛ ♥♦t✐♦♥ t❤❛t ✇✐•• ❛♣♣❡❛r ♥❛t✉r❛••② ✇❤❡♥ st✉❞②✐♥❣ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s✳ ❋✉rt❤❡r ♠♦✲ t✐✈❛t✐♦♥ ✐s ❣✐✈❡♥ ❜② ❛ r❡❝❡♥t ❝♦♥•❡❝t✉r❡ ♦❢ ▲❡✈✐t✐♥❛✱ ❙✉❦♦❝❤❡✈ ❛♥❞ )❛♥✐♥ ✭▲❙)✮✳ ❲❤❡♥ ✐♥✈❡st✐❣❛t✐♥❣ ❈✇✐❦❡• ❡st✐♠❛t❡s✱ t❤❡② st✉♠❜•❡❞ ✉♣♦♥ ❛ ❝♦♥❞✐t✐♦♥ ❢♦r ❛ s❡✲ q✉❡♥❝❡ s♣❛❝❡ t❤❛t ✇❡ ✇✐•• ❝❛•• r✐❣❤t✲✷✲♠♦♥♦t♦♥✐❝✐t② ❛♥❞ ✇❤✐❝❤ ✇✐•• ❜❡ ❞❡✜♥❡❞ •❛t❡r✳ ■♥ t❤♦s❡ t❡r♠s✱ t❤❡✐r ❝♦♥•❡❝t✉r❡ ❝❛♥ ❜❡ ❢♦r♠✉•❛t❡❞ ❛s ❢♦••♦✇s✿ ❈♦♥•❡❝t✉r❡ ✶✳✶✳✶ ✭▲❡✈✐t✐♥❛✱ ❙✉❦♦❝❤❡✈✱ )❛♥✐♥✮✳ ❆ s②♠♠❡tr✐❝ s❡q✉❡♥❝❡ s♣❛❝❡ E ✐s r✐❣❤t✲✷✲♠♦♥♦t♦♥❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ t❤❡r❡ ❡①✐sts p ∈ (0, 2) s✉❝❤ t❤❛t E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❢♦r t❤❡ ❝♦✉♣•❡ (ℓ p , ℓ 2 )✳ ✷✼ ◆❡✐t❤❡r ♦❢ t❤❡ t✇♦ ✐♠♣•✐❝❛t✐♦♥s ❛r❡ ❦♥♦✇♥✳ ■♥ t❤❡ ❇❛♥❛❝❤ ❝❛s❡ ❤♦✇❡✈❡r✱ t❤❡ ♣r♦❜•❡♠ ✇❛s ❛•♠♦st ❝♦♠♣•❡t❡•② s♦•✈❡❞ ❜② ❈✇✐❦❡• ❛♥❞ ◆✐❡•s♦♥✳ ■♥❞❡❡❞✱ t❤❡② s❤♦✇❡❞ t❤❛t ✐❢ E ❛ ❇❛♥❛❝❤ s❡q✉❡♥❝❡ s♣❛❝❡ ✇✐t❤ t❤❡ ❋❛t♦✉ ♣r♦♣❡rt② t❤❡♥ E ✐s •❡❢t✲✷✲♠♦♥♦t♦♥❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ ✐t ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❢♦r t❤❡ ❝♦✉♣•❡ (ℓ 1 , ℓ 2 )✳ ❙✉❜st❛♥t✐❛• ❞✐✣❝✉•t✐❡s ❛♣♣❡❛r ✇❤❡♥ ❝♦♥s✐❞❡r✐♥❣ q✉❛s✐✲❇❛♥❛❝❤ s❡q✉❡♥❝❡ s♣❛❝❡s✳ ❲❡ ❛r❡ ♠♦r❡ ✐♥t❡r❡st❡❞ ✐♥ ❢✉♥❝t✐♦♥ s♣❛❝❡s✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✐❢ t❤❡② ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ ♥❛t✉r❛•✱ ❛ ♥♦t✐♦♥ ✐♥tr♦❞✉❝❡❞ ❜② ❑❛•t♦♥ ❛♥❞ •❛t❡r ❞❡✜♥❡❞ ✐♥ t❤❡ t❡①t✱ ❛✇❡ ♦❜t❛✐♥✿ Pr♦♣♦s✐t✐♦♥ ✶✳✶✳✷✳ ▲❡t α ∈ (0, ∞] ❛♥❞ E ❛ ♥❛t✉r❛• s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡ ♦♥ (0, α) ✇✐t❤ t❤❡ ❋❛t♦✉ ♣r♦♣❡rt②✳ E ✐s r✐❣❤t✲✷✲♠♦♥♦t♦♥❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ t❤❡r❡ ❡①✲ ✐sts p ∈ (0, 2) s✉❝❤ t❤❛t E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❢♦r t❤❡ ❝♦✉♣•❡ (L p (0, α), L 2 (0, α))✳ ✶✳✷ ❙♦♠❡ r❡s✉•ts ❢r♦♠ ❛❜str❛❝t ✐♥t❡r♣♦•❛t✐♦♥ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ❣✐✈❡ ❛ ❜r✐❡❢ ✐♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ t❤❡♦r② ♦❢ ✐♥t❡r♣♦•❛t✐♦♥✳ ❆ ❞❡t❛✐•❡❞ ❡①♣♦s✐t✐♦♥ ♦❢ t❤✐s t♦♣✐❝ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ✈❛r✐♦✉s ❜♦♦❦s ❬✶(✳ ◆♦♥❡t❤❡•❡ss✱ ✇❡ ✇✐•• r❡❝❛•• s♦♠❡ ❞❡✜♥✐t✐♦♥s ❛♥❞ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ❜♦t❤ t♦ ♦❜t❛✐♥ ❛ ♠♦r❡ s❡•❢✲ ❝♦♥t❛✐♥❡❞ t❡①t ❛♥❞ t♦ ✜① s♦♠❡ ♣♦t❡♥t✐❛••② ♥♦♥✲st❛♥❞❛r❞ t❡r♠✐♥♦•♦❣②✳ ✶✳✷✳✶ ❋✐rst ❞❡✜♥✐t✐♦♥s ■♥t❡r♣♦•❛t✐♦♥ t❤❡♦r② ❛••♦✇s ♦♥ ♦♥❡ ❤❛♥❞ t♦ ❝♦♥str✉❝t ♥❡✇ s♣❛❝❡s ✇❤✐❝❤ •✐❡ ✧✐♥ ❜❡t✇❡❡♥✧ t✇♦ ❣✐✈❡♥ q✉❛s✐✲❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞ ♦♥ t❤❡ ♦t❤❡r t♦ ❞❡❞✉❝❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡s❡ ✧✐♥ ❜❡t✇❡❡♥✧ s♣❛❝❡s ❢r♦♠ t❤♦s❡ ♦❢ t❤❡ ❡♥❞✲s♣❛❝❡s✳ ■t ✐s ♦♥❡ ♦❢ t❤❡ ♠❛✐♥ t❡❝❤♥✐q✉❡s ❡♠♣•♦②❡❞ ✐♥ L p ✲s♣❛❝❡s t❤❡♦r②✳ ■♥❞❡❡❞✱ s✐♥❝❡ ❢♦r ❛♥② p < q < r ∈ (0, ∞]✱ L q ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ L p ❛♥❞ L r ✱ ♠❛♥② ✐♥❡q✉❛•✐t✐❡s ❢♦r L p ✲♥♦r♠s ❝❛♥ ❜❡ ❞❡❞✉❝❡❞ ❜② st✉❞②✐♥❣ ♦♥•② t❤❡ ❝❛s❡s ♦❢ L 1 ✱ L 2 ❛♥❞ L ∞ ✇❤✐❝❤ ❛r❡ ♦❢t❡♥ ♠♦r❡ tr❛❝t❛❜•❡✳ ❲❡ ✇✐•• s❛② t❤❛t ❛ ❝♦✉♣•❡ ♦❢ q✉❛s✐✲❇❛♥❛❝❤ s♣❛❝❡s (A, B) ✐s ❝♦♠♣❛t✐❜•❡ ✐❢ ✐t ✐s ❛ss♦❝✐❛t❡❞ t♦ ❛ t♦♣♦•♦❣✐❝❛• ✈❡❝t♦r s♣❛❝❡ C ❛♥❞ ❡♠❜❡❞❞✐♥❣s ♦❢ A ❛♥❞ B ✐♥t♦ C✳ ❚❤✐s ❡♥❛❜•❡s ✉s t♦ ❝♦♥s✐❞❡r t❤❡ s✉♠ A + B ❛♥❞ t❤❡ ✐♥t❡rs❡❝t✐♦♥ A ∩ B ♦❢ A ❛♥❞ B✳ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✶✳ ▲❡t (A, B) ❜❡ ❛ ❝♦♠♣❛t✐❜•❡ ❝♦✉♣•❡ ♦❢ q✉❛s✐✲❇❛♥❛❝❤ s♣❛❝❡s✳ ❲❡ s❛② t❤❛t ❛ q✉❛s✐✲❇❛♥❛❝❤ s♣❛❝❡ E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ✭✇✐t❤ ❝♦♥st❛♥t C✮ ❢♦r t❤✐s ❝♦✉♣•❡ ✐❢ A ∩ B ⊂ E ⊂ A + B ❛♥❞ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C > 0 s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r T : A + B → A + B ✇❤♦s❡ r❡str✐❝t✐♦♥ t♦ A ✭r❡s♣✳B✮ ✐s ❛ ❝♦♥tr❛❝t✐♦♥ ❢r♦♠ A t♦ A ✭r❡s♣✳ B t♦ B✮✱ T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ E t♦ E ✇✐t❤ ♥♦r♠ •❡ss t❤❛♥ C✳ E ✐s s❛✐❞ t♦ ❜❡ ❛♥ ❡①❛❝t ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ✐❢ C = 1✳ ❇♦t❤ ❢♦r ❡①♣•✐❝✐t ❝♦♥str✉❝t✐♦♥s ♦❢ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ✇✐t❤ t❤❡ r❡❛• ♠❡t❤♦❞ ❛♥❞ ❢♦r t❤❡ ❣❡♥❡r❛• t❤❡♦r② ♦❢ ✐♥t❡r♣♦•❛t✐♦♥✱ t❤❡ K✲❢✉♥❝t✐♦♥❛• ✐s ❛ ❢✉♥❞❛♠❡♥t❛• t♦♦•✳ ■t ✐s ❞❡✜♥❡❞ ❛s ❢♦••♦✇s✿ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✷✳ ▲❡t (A, B) ❜❡ ❛ ❝♦♠♣❛t✐❜•❡ ❝♦✉♣•❡ ♦❢ q✉❛s✐✲❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞ x ∈ A + B✳ ❋♦r ❛•• t > 0 ❞❡✜♥❡ t❤❡ K✲❢✉♥❝t✐♦♥❛• ♦❢ x ❜②✿ K t (x, A, B) = inf{ y A + t z B : y ∈ A, z ∈ B, y + z = x}. |❡♠❛r❦ ✶✳✷✳✸✳ ▲❡t (A, B) ❜❡ ❛ ❝♦♠♣❛t✐❜•❡ ❝♦✉♣•❡ ♦❢ ❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞ t > 0✳ ❚❤❡♥ A + B ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ q✉❛s✐✲♥♦r♠ . A+tB = K t (., A, B) ✐s ❛♥ ❡①❛❝t ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❢♦r t❤❡ ❝♦✉♣•❡ (A, B) t❤❛t ✇❡ ✇✐•• ❞❡♥♦t❡ ❜② A + tB✳ ■t ❡♥❛❜•❡s t♦ st❛t❡ ❛ s✐♠♣•❡ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ❛ s♣❛❝❡ t♦ ❜❡ ❛♥ ✐♥t❡r♣♦✲ •❛t✐♦♥ s♣❛❝❡✳ ❉❡✜♥✐t✐♦♥ ✶✳✷✳✹ ✭K✲♠♦♥♦t♦♥❡✮✳ ▲❡t (A, B) ❜❡ ❛ ❝♦♠♣❛t✐❜•❡ ❝♦✉♣•❡ ♦❢ q✉❛s✐✲ ❇❛♥❛❝❤ s♣❛❝❡s✳ ❲❡ s❛② t❤❛t ❛ q✉❛s✐✲❇❛♥❛❝❤ s♣❛❝❡ E ✐s K✲♠♦♥♦t♦♥❡ ❢♦r (A, B) ✐❢ A ∩ B ⊂ E ⊂ A + B ❛♥❞ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C s✉❝❤ t❤❛t ❢♦r ❛•• x ∈ E ❛♥❞ y ∈ A + B✱ ✈❡r✐❢②✐♥❣ ∀t > 0, K t (x, A, B) ≥ K t (y, A, B), t❤❡♥ y ∈ E ❛♥❞ y E ≤ C x E . ❚❤❡ ❢♦••♦✇✐♥❣ ❢❛❝t ✐s ✇❡••✲❦♥♦✇♥✱ ✇❡ ♣r♦✈❡ ✐t ❤❡r❡ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ q✉❛s✐✲ ❇❛♥❛❝❤ s♣❛❝❡s ❢♦r ❝♦♠♣•❡t✐♦♥✳ Pr♦♣♦s✐t✐♦♥ ✶✳✷✳✺✳ ■❢ E ✐s K✲♠♦♥♦t♦♥❡ ✇✐t❤ ❝♦♥st❛♥t C ❢♦r t❤❡ ❝♦✉♣•❡ (A, B) t❤❡♥ E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ A ❛♥❞ B ✇✐t❤ t❤❡ s❛♠❡ ❝♦♥st❛♥t✳ Pr♦♦❢✳ ▲❡t T : A + B → A + B ❜❡ ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r s✉❝❤ t❤❛t ✐ts r❡str✐❝t✐♦♥ t♦ A ✭r❡s♣✳ B✮ ✐s ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r ✇✐t❤ ♥♦r♠ 1 ♦♥ A ✭r❡s♣✳ B✮✳ ▲❡t x ∈ E ❛♥❞ •❡t y = T x✱ y ∈ A + B✳ ▲❡t t > 0✱ ❛s ♥♦t❡❞ ✐♥ r❡♠❛r❦ ✶✳✷✳✸✱ A + tB ✐s ❛♥ ❡①❛❝t ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❢♦r t❤❡ ❝♦✉♣•❡ (A, B)✳ ❚❤✐s ♠❡❛♥s t❤❛t ❢♦r ❛•• t > 0✱ K t (x, A, B) ≥ K t (y, A, B)✳ ❍❡♥❝❡✱ ❜② K✲♠♦♥♦t♦♥✐❝✐t② ♦❢ E✱ y ∈ E ❛♥❞ y E ≤ C x E ✳ ❙♦ T ❞❡✜♥❡s ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r ♦❢ ♥♦r♠ •❡ss t❤❛♥ C ♦♥ E✳ ■❢✱ r❡❝✐♣r♦❝❛••②✱ ❡✈❡r② ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ♦❢ ❛ ❝♦✉♣•❡ (A, B) ✐s K✲♠♦♥♦t♦♥❡✱ (A, B) ✐s s❛✐❞ t♦ ❜❡ ❛ ❈❛•❞❡ró♥ ❝♦✉♣•❡✳ ■❢ t❤❡ ✉♥✐t ❜❛•• ♦❢ A ❛♥❞ t❤❡ ✉♥✐t ❜❛•• ♦❢ B ❛r❡ ❝•♦s❡❞ ✐♥ A + B ✭❢♦r ✐ts ♥❛t✉r❛• ♥♦r♠ ✐✳❡ K 1 (., A, B)✮ t❤❡♥ t❤❡ ❝♦✉♣•❡ (A, B) ✐s s❛✐❞ t♦ ❜❡ ❛ •❛❣•✐❛r❞♦ ❝♦✉♣•❡ ♦r •❛❣•✐❛r❞♦ ❝♦♠♣•❡t❡✳ ✶✳✸ ❑✲♠♦♥♦t♦♥✐❝✐t② ❢♦r t❤❡ ❝♦✉♣•❡ (L p , L q ) ✶✳✸✳✶ ❉❡✜♥✐t✐♦♥s ❛♥❞ ♦✈❡r✈✐❡✇ ♦❢ t❤❡ •✐t❡r❛t✉r❡ ❲❡ st❛rt ❜② ✐♥tr♦❞✉❝✐♥❣ s♦♠❡ ♥♦t❛t✐♦♥s✳ ❋✐① α ∈ (0, ∞]✳ ❉❡♥♦t❡ ❜② L 0 (0, α) t❤❡ s❡t ♦❢ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥s f ♦♥ (0, α)✱ s✉❝❤ t❤❛t 1 |f |>t ✐s ✜♥✐t❡ ❢♦r s♦♠❡ t ∈ R✳ ❋♦r f ∈ L 0 (O, α)✱ ❞❡♥♦t❡ ❜② f * t❤❡ ♥♦♥✲✐♥❝r❡❛s✐♥❣ r❡❛r❛♥❣❡♠❡♥t ♦❢ f ✳ ❉❡✜♥✐t✐♦♥ ✶✳✸✳✶ ✭❙②♠♠❡tr✐❝ s♣❛❝❡✮✳ ❲❡ ❝❛•• q✉❛s✐✲❇❛♥❛❝❤ s②♠♠❡tr✐❝ ❢✉♥❝✲ t✐♦♥ s♣❛❝❡ ❛ ♥♦♥③❡r♦ s✉❜s♣❛❝❡ ♦❢ L 0 (0, α) ✇❤✐❝❤ ✐s r❡❛rr❛♥❣❡♠❡♥t ✐♥✈❛r✐❛♥t ✭t❤❡ q✉❛s✐✲♥♦r♠ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦♥•② ❞❡♣❡♥❞s ♦♥ ✐ts ❞✐str✐❜✉t✐♦♥✮ ❛♥❞ ❡q✉✐♣♣❡❞ ✇✐t❤ ❛♥ ✐♥❝r❡❛s✐♥❣ q✉❛s✐✲♥♦r♠✳ ❙✐♠✐•❛r•②✱ ❛ q✉❛s✐✲❇❛♥❛❝❤ s②♠♠❡tr✐❝ s❡q✉❡♥❝❡ s♣❛❝❡ ✐s ❛ s✉❜s♣❛❝❡ ♦❢ ℓ ∞ ❡♥•♦②✐♥❣ t❤❡ s❛♠❡ ♣r♦♣❡rt✐❡s✳ ❆ s②♠♠❡tr✐❝ s♣❛❝❡ E ✐s s❛✐❞ t♦ ❤❛✈❡ t❤❡ ❋❛t♦✉ ♣r♦♣❡rt② ✐❢ ❢♦r ❡✈❡r② ✐♥❝r❡❛s✐♥❣ ♥❡t (f n ) n∈I ♦❢ ❡•❡♠❡♥ts ♦❢ E s✉❝❤ t❤❛t ( f n E ) n∈I ✐s ❜♦✉♥❞❡❞ ❛♥❞ f n ↑ f ❛✳❡✳✱ t❤❡♥ f ∈ E ❛♥❞ f n E ↑ f E ✳ ❆♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ s②♠♠❡tr✐❝ s♣❛❝❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✸✷(✳ |❡❝❛•• t❤❛t E ✐s s❛✐❞ t♦ ❜❡ p✲❝♦♥✈❡① ✇✐t❤ ❝♦♥st❛♥t C ✐❢ ❢♦r ❛•• n ∈ N ❛♥❞ (x i ) i≤n ∈ E n ✱ n i=1 |x i | p 1/p E ≤ C n i=1 x i p E 1/p . ❙✐♠✐•❛r•②✱ E ✐s s❛✐❞ t♦ ❜❡ q✲❝♦♥❝❛✈❡ ✇✐t❤ ❝♦♥st❛♥t C ✐❢ ❢♦r ❛•• n ∈ N ❛♥❞ (x i ) i≤n ∈ E n ✱ n i=1 x i q E 1/q ≤ C n i=1 |x i | q 1/q E . ❆ s②♠♠❡tr✐❝ s♣❛❝❡ ✐s s❛✐❞ t♦ ❜❡ ♥❛t✉r❛• ✐❢ ✐t ✐s p✲❝♦♥✈❡① ❢♦r s♦♠❡ p✳ ❚❤✐s ✐♥❝•✉❞❡s ▲♦r❡♥t③ s♣❛❝❡s ❛♥❞ ❖r•✐❝③ s♣❛❝❡s✳ ❚❤❡r❡ ❡①✐sts s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡s ✇❤✐❝❤ ❛r❡ ♥♦t ♥❛t✉r❛• ❜✉t ❝♦♥str✉❝t✐♥❣ ♦♥❡ ✐s ♥♦t ❡❛s②✱ s❡❡ ❢♦r ❡①❛♠♣•❡ ❬✷✾( ❢♦r ❛♥ ✐❞❡❛ ♦❢ ❤♦✇ t♦ ❞♦ ✐t✳ |❡♠❛r❦ t❤❛t ❛•• ❇❛♥❛❝❤ s♣❛❝❡s ❛r❡ ♥❛t✉r❛• s✐♥❝❡ t❤❡② ❛r❡ 1✲❝♦♥✈❡①✳ ❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡s ✇❤✐❝❤ ❛r❡ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ♦❢ t❤❡ ❝♦✉♣•❡ (L p , L q ) r❡•✐❡ ♦♥ ✐ts K✲❢✉♥❝t✐♦♥❛•✳ ■t ✇❛s ❝♦♠♣✉t❡❞ ❜② ❑ré❡ ❛♥❞ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✷✶(✳ Pr♦♣♦s✐t✐♦♥ ✶✳✸✳✷✳ ▲❡t p < q ∈ (0, ∞]✱ t❤❡♥ ❢♦r ❛•• f ∈ L p (0, α) + L q (0, α)✿ K t (f, L p , L q ) ≈ t θ 0 (f * ) p 1/p + t ∞ t θ (f * ) q 1/q . ✭✶✳✶✮ ✇✐t❤ θ = (p -1 -q -1 ) -1 ✳ ■❢ q = ∞✱ t❤❡ ❢♦r♠✉•❛ ✐s t♦ ❜❡ ✉♥❞❡rst♦♦❞ ❛s ❢♦••♦✇s✿ K t (f, L p , L ∞ ) ≈ Ap t p 0 (f * ) p 1/p . ✭✶✳✷✮ ■♥ ✈✐❡✇ ♦❢ t❤✐s s❡❝♦♥❞ ❢♦r♠✉•❛✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣ ❞❡✜♥✐t✐♦♥s✳ ❉❡✜♥✐t✐♦♥ ✶✳✸✳✸ ✭▼❛•♦r✐③❛t✐♦♥✮✳ ❋♦r t✇♦ ❡•❡♠❡♥ts x, y ∈ L 0 (0, α)✱ ✇❡ ✇r✐t❡ x ≻ y ❛♥❞ ✇❡ s❛② t❤❛t x •❡❢t✲♠❛•♦r✐③❡s y ✐❢ ❛♥❞ ♦♥•② ✐❢ ❢♦r ❛•• t > 0✱ t 0 x * ≥ t 0 y * ✳ ❙✐♠✐•❛r•②✱ ✇❡ ✇r✐t❡ x ⊲ y ❛♥❞ ✇❡ s❛② t❤❛t x r✐❣❤t✲♠❛•♦r✐③❡s y ✐❢ ❛♥❞ ♦♥•② ✐❢ ❢♦r ❛•• t > 0✱ ∞ t x * ≥ ∞ t y * ✳ ✶✳✸✳ ❑✲▼❖◆❖❚❖◆■❈■❚( ❋❖| ❚❍❊ ❈❖❯P▲❊ (L P , L Q ) ✸✶ ❚❤❡s❡ ♥♦t✐♦♥s ❛•r❡❛❞② ❛♣♣❡❛r ✐♥ t❤❡ •✐t❡r❛t✉r❡ ❬✾(✳ ❚❤❡ ♦r❞❡r ❣✐✈❡♥ ❜② •❡❢t✲ ♠❛•♦r✐③❛t✐♦♥ ✐s ♣❛rt✐❝✉•❛r•② ✉s❡❢✉• s✐♥❝❡ ❢♦r t✇♦ ❢✉♥❝t✐♦♥s f, g✱ t❤❡r❡ ❡①✐sts ❛♥ ✐♥t❡❣r❛• ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈❡ s✉❜✉♥✐t❛• ♦♣❡r❛t♦r T ♦♥ L 1 (0, ∞) + L ∞ (0, ∞) s✉❝❤ t❤❛t T f ≥ g ✐❢ ❛♥❞ ♦♥•② ✐❢ f ≻ g ❬✻(✳ ■♥ ♠❛♥② ❝❛s❡s✱ t❤❡ ❝♦✉♣•❡ (L p , L q ) ✐s ❦♥♦✇♥ t♦ ❜❡ ❛ ❈❛•❞❡ró♥ ❝♦✉♣•❡✳ ❚❤✐s ✐s s✉♠♠❛r✐③❡❞ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ✭❬✸✺(✱❬✺✼(✱❬✾(✱❬✶✵(✮✳ ❚❤❡♦r❡♠ ✶✳✸✳✹✳ ❚❤❡ ❢♦••♦✇✐♥❣ ❝♦✉♣•❡s ♦❢ q✉❛s✐✲❇❛♥❛❝❤ s♣❛❝❡s ❛r❡ ❈❛•❞❡ró♥ ❝♦✉♣•❡s✿ • ✭▲♦r❡♥t③✲❙❤✐♠♦❣❛❦✐✮ p ≥ 1✱ (L p (0, α), L ∞ (0, α)) ❛♥❞ (L 1 (0, α), L p (0, α))✱ • ✭❙♣❛rr✮ p, q ≥ 1✱ (L p (Ω), L q (Ω)) ❢♦r ❛♥② σ✲✜♥✐t❡ ♠❡❛s✉r❡ s♣❛❝❡ Ω✳ • ✭❙♣❛rr✱ ❈✇✐❦❡•✮ p, q ∈ (0, ∞)✱ (L p (0, α), L q (0, α))✱ • ✭❈✇✐❦❡•✮ p ∈ (0, ∞]✱ (ℓ p , ℓ ∞ )✳ ✶✳✸✳✷ ❚❤❡ ❝♦✉♣•❡ (L p (0, α), L ∞ (0, α))✱ p < 1 ❋♦r t❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤✐s s❡❝t✐♦♥✱ ✜① p ∈ (0, ∞)✳ ❚❤❡♦r❡♠ ✶✳✸✳✺✳ ❊✈❡r② ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ♦❢ t❤❡ ❝♦✉♣•❡ (L p (0, α), L ∞ (0, α)) ✐s K✲♠♦♥♦t♦♥❡✳ ❲❡ ✇✐•• ❞❡❞✉❝❡ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡ ❢r♦♠ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✿ Pr♦♣♦s✐t✐♦♥ ✶✳✸✳✻✳ ▲❡t f, g ∈ L p (0, α) + L ∞ (0, α) ❛♥❞ d > 0✳ ❙✉♣♣♦s❡ t❤❛t f ❛♥❞ g ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡✱ ♥♦♥✲✐♥❝r❡❛s✐♥❣✱ •❡❢t✲❝♦♥t✐♥✉♦✉s✱ t❛❦❡ ✈❛•✉❡s ✐♥ D = {d n } n∈Z ∪ {0} ❛♥❞ f p ≻ g p ✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ♦♣❡r❛t♦r T : L p (0, α) + L ∞ (0, α) → L p (0, α) + L ∞ (0, α) ❛♥❞ ❛ ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ h s✉❝❤ t❤❛t✿ • h ≤ 2 1/p f • T h * = g✱ • t❤❡ r❡str✐❝t✐♦♥s ♦❢ T ♦♥ L p (0, α) ❛♥❞ L ∞ (0, α) ❛r❡ ❝♦♥tr❛❝t✐♦♥s✳ Pr♦♦❢✳ ❉❡✜♥❡ h := 2 1/p f 1 2f p ≥g p ✳ ▲❡t t > 0✱ •❡t ✉s s❤♦✇ t❤❛t t 0 h p ≥ t 0 g p ✳ ◆♦t❡ t❤❛t g p > 2f p ⇔ g p < 2(g p -f p )✳ t 0 h p - t 0 g p = t 0 1 2f p ≥g p (2f p -g p ) -1 2f p <g p g p ≥ t 0 1 2f p ≥g p 2(f p -g p ) -1 2f p <g p 2(g p -f p ) = 2 t 0 f p -g p ≥ 0. ❉❡♥♦t❡ X := g p ∈ (0, ∞]✳ ❉❡✜♥❡✿ ✸✷ ❈❍❆P❚❊| ✶✳ ■◆❚❊|P❖▲❆❚■❖◆ ❙P❆❈❊❙ ❇❊❚❲❊❊◆ L P ✲❙P❆❈❊❙ H : t → t 0 h p ❛♥❞ G : t → t 0 g p ✳ ▲❡t x ∈ [0, X)✱ ♥♦t❡ t❤❛t G -1 (x) ✐s ✇❡••✲❞❡✜♥❡❞ s✐♥❝❡ G ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ✐♥❝r❡❛s✐♥❣ ♦♥ [0, β]✱ t❤❡ s✉♣♣♦rt ♦❢ g✳ ❉❡✜♥❡ ❛•s♦ H -1 (x) = min H -1 ({x}) ✭H ✐s ❝♦♥t✐♥✉♦✉s✮✳ ◆♦t❡ t❤❛t h(H -1 (x)) = 0✳ ■♥❞❡❡❞✱ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ t x := H -1 (x)✱ ❢♦r ❛•• ε > 0✱ tx tx-ε h = 0 s♦ t❤❡r❡ ❡①✐sts ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ (t n ) ❝♦♥✈❡r❣✐♥❣ t♦ t x ❛♥❞ s✉❝❤ t❤❛t h(t n ) = 0✳ ❚❤✐s ♠❡❛♥s t❤❛t 2f (t n ) ≥ g(t n )✳ ❯s✐♥❣ t❤❡ ❢❛❝t t❤❛t f ✐s •❡❢t✲❝♦♥t✐♥✉♦✉s ❛♥❞ g ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣✱ ✇❡ ♦❜t❛✐♥ 2f (t x ) ≥ g(t x )✳ ❆♥❞ s✐♥❝❡ ✇❡ s❤♦✇❡❞ t❤❛t H ≥ G✱ H -1 (x) ≤ G -1 (x)✱ ❤❡♥❝❡ g p (H -1 (x)) ≥ g p (G -1 (x))✳ ■♥ ❝♦♥❝•✉s✐♦♥✿ h p (H -1 (x)) ≥ g p (G -1 (x)). ✭✶✳✸✮ ❈♦♥s✐❞❡r ♥♦✇ h * ❛♥❞ H * : t → t 0 (h * ) p ✳ ❇② ❝♦♥str✉❝t✐♦♥✱ h r❡str✐❝t❡❞ t♦ {h > 0} ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣ ❛♥❞ h * ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣ t♦♦✳ ❙✐♥❝❡ h ❛♥❞ h * ❛•s♦ ❤❛✈❡ ❛ ❝♦✉♥t❛❜•❡ r❛♥❣❡✱ t❤❡② ❛r❡ ♦❢ t❤❡ ❢♦r♠ i∈N a i 1 Ii ✇❤❡r❡ t❤❡ I i ❛r❡ ✐♥t❡r✈❛•s✳ ▲❡t t ✐♥ t❤❡ s✉♣♣♦rt ♦❢ h s♦ t❤❛t t ✐s ♥♦t ❛♥ ❡♥❞ ♣♦✐♥t ♦❢ ♦♥❡ ♦❢ t❤♦s❡ ✐♥t❡r✈❛•s t❤❡♥ s✐♥❝❡ h ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣✿ h>h(t) h < t 0 h < h≥h(t) h. ❙✐♠✐•❛r•②✱ ✐❢ t ′ ✐s ✐♥ t❤❡ s✉♣♣♦rt ♦❢ h * ❛♥❞ ♥♦t ❛t ❛♥ ❡♥❞ ♣♦✐♥t✱ h * >h * (t ′ ) h * < t ′ 0 h * < h * ≥h * (t ′ ) h * . ❙✐♥❝❡ h ❛♥❞ h * ❤❛✈❡ t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥✱ t❤✐s ♠❡❛♥s t❤❛t ❢♦r ❛•• s > 0✱ h>s h = h * >s h * ❛♥❞ h≥s h = h * ≥s h * ✳ ❇② t❤❡ ♣r❡✈✐♦✉s ✐♥❡q✉❛•✐t✐❡s✱ ✇❡ ❞❡❞✉❝❡ t❤❛t ❡①❝❡♣t ❢♦r ❛ ❝♦✉♥t❛❜•❡ ♥✉♠❜❡r ♦❢ t ❛♥❞ t ′ ✱ ✐❢ H(t) = H * (t ′ ) t❤❡♥ h(t) = h * (t ′ )✳ ❙✐♥❝❡ H * ✐s str✐❝t•② ✐♥❝r❡❛s✐♥❣ ♦♥ H -1 * ((0, X))✱ t❤✐s ✐♠♣•✐❡s ❜② ✭✶✳✸✮ t❤❛t ❢♦r ❛•• x ∈ (0, X)✱ ❡①❝❡♣t ♠❛②❜❡ ❛ ❝♦✉♥t❛❜•❡ ♥✉♠❜❡r✱ h * (H -1 * (x)) = h(H -1 (x)) ≥ g(G -1 (x)). ✭✶✳✹✮ ▲❡t φ = H -1 * • G ♦♥ (0, β)✳ |❡♠❛r❦ t❤❛t s✐♥❝❡ h * ❛♥❞ g ❤❛✈❡ r❛♥❣❡ ✐♥ 2 1/p D ❛♥❞ D r❡s♣❡❝t✐✈❡•②✱ H * ❛♥❞ G ❛r❡ ♣✐❡❝❡✇✐s❡ ❛✣♥❡ ♦♥ ❡✈❡r② ❝♦♠♣❛❝t ♦❢ (0, β)✳ ❙✐♥❝❡ H * • φ = G✱ ❜② ❞❡r✐✈❛t✐♥❣✱ ❢♦r ❛•• t ∈ (0, α)✱ ❡①❝❡♣t ♠❛②❜❡ ❛ ❝♦✉♥t❛❜•❡ ♥✉♠❜❡r✱ φ ′ (t)(h * • φ) p (t) = g p (t). ✭✶✳✺✮ ❍❡♥❝❡ ❞❡✜♥❡ T : L p (0, α)+L ∞ (0, α) → L p (0, α)+L ∞ (0, α) ❜② T (u) = (φ ′1/p •u• φ) •1 (0,β) ✳ ❇② t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s ❢♦r♠✉•❛✱ T ✐s ❛ ❝♦♥tr❛❝t✐♦♥ ♦♥ L p ✳ ■♥❞❡❡❞✱ ♦♥❡ ❝❛♥ ❛♣♣•② t❤❡ ❢♦r♠✉•❛ ♦♥ ❛♥② ❝♦♠♣❛❝t ♦❢ (0, β) ❛♥❞ ♦❜t❛✐♥ t❤❡ p✲♥♦r♠ ❛s ❛ •✐♠✐t✳ ▲❡t t ∈ (0, α) ❛♥❞ ✇r✐t❡ t = G -1 (x)✱ t❤❡♥ ❜② ✭✶✳✹✮✿ φ ′ (t) = (H -1 * • G) ′ (t) = g p (t) (h * ) p (H -1 * (G(t)) = g(G -1 (x)) h * (H -1 * (x)) p ≤ 1, ✶✳✹✳ P ✲▼❖◆❖❚❖◆■❈■❚( ✸✸ ❡①❝❡♣t ❢♦r ❛ ❝♦✉♥t❛❜•❡ ♥✉♠❜❡r ♦❢ ✈❛•✉❡s ♦❢ t✳ ❙♦ T ✐s ❛•s♦ ❛ ❝♦♥tr❛❝t✐♦♥ ♦♥ L ∞ (0, α)✳ ❋✐♥❛••②✱ ❜② ✭✶✳✺✮✱ T h * = g✳ Pr♦♦❢ ♦❢ t❤❡♦r❡♠ ✶✳✸✳✺✳ ▲❡t E ❜❡ ❛ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❢♦r t❤❡ ❝♦✉♣•❡ (L p (0, α), L ∞ (0, α)) ✇✐t❤ ❝♦♥st❛♥t C✳ ▲❡t f ∈ E ❛♥❞ g ∈ L p + L ∞ s✉❝❤ t❤❛t ❢♦r ❛•• t > 0✱ K t (f, L p (0, α), L ∞ (0, α) ≥ K t (g, L p (0, α), L ∞ (0, α)✳ ❇② ✭✶✳✷✮✱ ✇❡ ❝❛♥ r❡♣•❛❝❡ t❤✐s ❝♦♥❞✐t✐♦♥ ❜② f p ≻ g p ✳ ▲❡t d ∈ (1, ∞) ❛♥❞ ❞❡✜♥❡ f 2 = d ⌊log d (f * )⌋+1 ❛♥❞ g 2 = d ⌊log d (g * )⌋ ✳ ❲❡ ❤❛✈❡ f * ≤ f 2 ≤ df * ✱ g * ≤ dg 2 ≤ dg * ✱ ❤❡♥❝❡ f p 2 ≻ g p 2 ✳ ❋✉rt❤❡r♠♦r❡✱ f 2 ❛♥❞ g 2 t❛❦❡ ✈❛•✉❡s ✐♥ D = {d n } n∈Z ∪ {0}✳ ❚❛❦❡ h ❛♥❞ T ❣✐✈❡♥ ❜② ♣r♦♣♦s✐t✐♦♥ ✶✳✸✳✻✳ ❙✐♥❝❡ h ≤ 2 1/p f 2 ✱ h ∈ E s♦ ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡✱ T h = g 2 ∈ E ❛♥❞ ✜♥❛••② g ∈ E✳ ❋✉rt❤❡r♠♦r❡✿ g E = g * E ≤ d g 2 E ≤ dC h E ≤ dC2 1/p f 2 E ≤ d 2 C2 1/p f * E = d 2 C2 1/p f E . ❚❤✐s ✐s tr✉❡ ❢♦r ❛♥② d ∈ (1, ∞) s♦✱ g E ≤ C2 1/p f E ✳ ✶✳✹ p✲♠♦♥♦t♦♥✐❝✐t② ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥s ♦❢ •❡❢t✲p✲♠♦♥♦t♦♥✐❝✐t② ❛♥❞ r✐❣❤t✲p✲ ♠♦♥♦t♦♥✐❝✐t②✳ ❚❤✐s ❛••♦✇s ✉s t♦ ✜♥❞ ❛♥ ❡q✉✐✈❛•❡♥t ❢♦r♠✉•❛t✐♦♥ ❢♦r t❤❡ K✲ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ ❝♦✉♣•❡ (L p (0, α), L q (0, α))✳ ❉❡✜♥✐t✐♦♥ ✶✳✹✳✶ ✭p✲♠♦♥♦t♦♥✐❝✐t②✮✳ ❆ s②♠♠❡tr✐❝ q✉❛s✐✲❇❛♥❛❝❤ s♣❛❝❡ E ✐s s❛✐❞ t♦ ❜❡ •❡❢t✲p✲♠♦♥♦t♦♥❡ ✐❢ t❤❡r❡ ❡①✐sts C > 0 s✉❝❤ t❤❛t ❢♦r ❛•• f ∈ E ❛♥❞ g ∈ L 0 (0, α)✱ f p ≻ g p ⇒ g ∈ E, g E ≤ C f E . ■t ✐s s❛✐❞ t♦ ❜❡ r✐❣❤t✲p✲♠♦♥♦t♦♥❡ ✐❢ t❤❡r❡ ❡①✐sts C > 0 s✉❝❤ t❤❛t ❢♦r ❛•• f ∈ E ❛♥❞ g ∈ L 0 (0, α)✱ f p ⊲ g p ⇒ g ∈ E, g E ≤ C f E . |❡♠❛r❦ t❤❛t ❜② ✭✶✳✶✮✱ •❡❢t✲p✲♠♦♥♦t♦♥✐❝✐t② ✐s ✭❛•♠♦st✮ ❛ ♣❛rt✐❝✉•❛r ❝❛s❡ ♦❢ K✲ ♠♦♥♦t♦♥✐❝✐t② ✐✳❡✳ K✲♠♦♥♦t♦♥✐❝✐t② ❢♦r t❤❡ ❝♦✉♣•❡ (L p , L ∞ ) ✭t❤❡ ♦♥•② ❞✐✛❡r❡♥❝❡ •✐❡s ✐♥ t❤❡ ❢❛❝t t❤❛t ✇❡ ❞♦ ♥♦t s✉♣♣♦s❡ E t♦ ❜❡ ❛♥ ✐♥t❡r♠✐❞✐❛t❡ s♣❛❝❡✮✳ ❲❡ ✇✐•• ❞❡♥♦t❡ ❜② F t❤❡ s❡t ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ ❞②❛❞✐❝ st❡♣ ❢✉♥❝t✐♦♥s✿ F := f ∈ L ∞ (0, ∞) : ∃n ∈ N, ∃N ∈ N, ∃a ∈ (R + ) N , f = N i=1 a i 1 [(i-1)2 -n ,i2 -n ) . ◆♦t❡ t❤❛t t❤✐s s♣❛❝❡ ✇✐•• ❛•s♦ ❜❡ ✉s❡❢✉• •❛t❡r ✐♥ t❤✐s t❡①t✳ ▲❡♠♠❛ ✶✳✹✳✷✳ ▲❡t f, g ∈ F ❛♥❞ p ∈ (0, ∞)✳ • ■❢ f p ≻ g p t❤❡♥ t❤❡r❡ ❡①✐sts h ∈ F s✉❝❤ t❤❛t h ≥ g✱ f p ≻ h p ❛♥❞ f p = h p ✳ • ■❢ f p ⊲ g p t❤❡♥ t❤❡r❡ ❡①✐sts h ∈ F s✉❝❤ t❤❛t h ≥ g✱ h p ≻ f p ❛♥❞ f p = h p ✳ Pr♦♦❢✳ ❲❡ ❜❡•✐❡✈❡ t❤❛t ✐t ✐s ♥♦t ❞✐✣❝✉•t t♦ ❝♦♥✈✐♥❝❡ ♦♥❡s❡•❢ t❤❛t t❤✐s •❡♠♠❛ ✐s tr✉❡✳ ❍♦✇❡✈❡r✱ ❢♦r ❝♦♠♣•❡t❡♥❡ss✱ ✇❡ ❝♦♥str✉❝t ❡①♣•✐❝✐t ❡①♣r❡ss✐♦♥s ❢♦r h✳ ❲r✐t❡✿ f = N i=1 a i 1 (i-1)ε,iε) ❛♥❞ g = M i=1 b i 1 [(i-1)ε,iε) ✳ ❇② ❝❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ t❤❡ a i ❛♥❞ b i ✱ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t f ❛♥❞ g ❛r❡ ♥♦♥✲ ✐♥❝r❡❛s✐♥❣ ❛♥❞ r❡♣•❛❝✐♥❣ f ❜② f p ❛♥❞ g ❜② g p ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t p = 1✳ ❙✉♣♣♦s❡ ❛•s♦ t❤❛t ♥♦♥❡ ♦❢ t❤❡ a i ♥♦r b i ❛r❡ 0 ❙✉♣♣♦s❡ t❤❛t f ≻ g✳ ▲❡t δ = min(a N , b M ) ❛♥❞ D = f 1 -g 1 ✳ ❈❤♦♦s❡ n ∈ N s✉❝❤ t❤❛t εnδ > D ❛♥❞ •❡t b = D/εn✳ ❉❡✜♥❡ h = M +n i=1 c i 1 [(i-1)ε,iε) ✇✐t❤ c i = b i ❢♦r i ≤ M ❛♥❞ c i = b ❢♦r i ≥ M +1✳ ❙✐♥❝❡ b < δ✱ h ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣✳ h 1 = ε M +n i=0 c i = b 1 + nεb = b 1 + D = f 1 . ❚♦ ♣r♦✈❡ t❤❛t f ≻ g ❞❡✜♥❡✿ F : t → t 0 f ✱ G : t → t 0 g ❛♥❞ H : t → t 0 h✳ ❋♦r t ≤ M ε✱ F (t) ≥ G(t) = H(t)✳ ❙✐♥❝❡ F ❛♥❞ H ❛r❡ ❛✣♥❡ ♦♥ ✐♥t❡r✈❛•s ♦❢ t❤❡ ❢♦r♠ [iε, (i + 1)ε]✱i ∈ N✱ ✐t s✉✣❝❡s t♦ ♣r♦✈❡ t❤❛t F (iε) ≥ H(iε) ❢♦r ❛•• i ∈ N✳ ▲❡t ✉s ❛r❣✉❡ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ i✳ ❲❡ ❝❛♥ ✐♥✐t✐❛•✐s❡ ❛t i = M ✳ ▲❡t i ≥ M ❛♥❞ s✉♣♣♦s❡ t❤❛t F (iε) ≥ G(iε) t❤❡♥ t❤❡② ❛r❡ t✇♦ ♣♦ss✐❜✐•✐t✐❡s✿ ❡✐t❤❡r a i = 0 t❤❡♥ F ((i + 1)ε) = f 1 = h 1 ≥ H((i + 1)ε) ♦r a i = 0 t❤❡♥ ❜② ❤②♣♦t❤❡s✐s ♦♥ b✱ a i ≥ b ❛♥❞ F ((i + 1)ε) = F (iε) + a i ε ≥ G(iε) + b = G((i + 1)ε)✳ ❙✉♣♣♦s❡ ♥♦✇ t❤❛t f ⊲ g✳ ❉❡✜♥❡ h = N i=1 c i 1 [(i-1)ε,iε) ✳ ❲✐t❤ c i = b i ❢♦r ❛•• i ≥ 2 ❛♥❞ c 1 := N i=1 a i - N i=2 b i ✐s ❝❤♦s❡♥ s♦ t❤❛t N i=0 a i = N i=0 b i ✳ ❚♦ ✈❡r✐❢② t❤❡ ♠❛•♦r✐③❛t✐♦♥ ❝♦♥❞✐t✐♦♥✱ ❞❡✜♥❡✿ F : t → ∞ t f ✱ G : t → ∞ t g ❛♥❞ H : t → t 0 h✳ ❋♦r t ≥ ε✱ F (t) ≥ G(t) = H(t)✳ ❋✉rt❤❡r♠♦r❡✱ F (0) = H(0) ❜② ❝♦♥str✉❝t✐♦♥✱ F (ε) ≥ H(ε) ❛♥❞ F ❛♥❞ G ❛r❡ ❛✣♥❡ ♦♥ [0, ε] s♦ F (t) ≥ H(t) ❢♦r ❛•• t ∈ [0, ε]✳ ❚❤✐s ❝♦♠♣•❡t❡s t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ h✳ ◆♦t❡ t❤❛t ❢♦r ❛•• t > 0✿ t 0 h = h 2 -H(t) ≥ f 1 -F (t) = t 0 f. ❍❡♥❝❡✱ h ≻ f ✳ ✶✳✹✳✶ ❙✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r p✲♠♦♥♦t♦♥✐❝✐t② ■♥ t❤✐s s✉❜s❡❝t✐♦♥ ✇❡ ♣r♦✈❡ s♦♠❡ •❡♠♠❛s •✐♥❦✐♥❣ p✲❝♦♥✈❡①✐t② ✭r❡s♣✳ q✲❝♦♥❝❛✈✐t②✮ ❛♥❞ •❡❢t✲p✲♠♦♥♦t♦♥✐❝✐t② ✭r❡s♣✳ r✐❣❤t✲q✲♠♦♥♦t♦♥✐❝✐t②✮✳ ❋✐rst✱ ✇❡ s❤♦✇ t❤❛t p✲ ❝♦♥✈❡①✐t② ✐♠♣•✐❡s s♦♠❡ ✇❡❛❦ ❢♦r♠ ♦❢ t❤❡ •❡❢t✲p✲♠♦♥♦t♦♥✐❝✐t②✳ ❚❤❡♥✱ ✇❡ s❤♦✇ t❤❛t ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t E ❤❛s t❤❡ ❋❛t♦✉ ♣r♦♣❡rt②✱ t❤✐s ✇❡❛❦ ❢♦r♠ ♦❢ •❡❢t✲p✲♠♦♥♦t♦♥✐❝✐t② ✐s✱ ✐♥ ❢❛❝t✱ ❡q✉✐✈❛•❡♥t t♦ •❡❢t✲p✲♠♦♥♦t♦♥✐❝✐t② ❜② ❛ s✐♠♣•❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛r❣✉♠❡♥t✳ ❈♦♥✈❡①✐t② ✐♠♣•✐❡s •❡❢t✲♠♦♥♦t♦♥✐❝✐t② ❖✉r ✜rst •❡♠♠❛ ✐s ❛ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ❢♦r♠ ♦❢ t❤❡ ❙❝❤✉r✲❍♦r♥ t❤❡♦r❡♠✳ ▲❡♠♠❛ ✶✳✹✳✸✳ ▲❡t E ❜❡ ❛ q✉❛s✐✲❇❛♥❛❝❤ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡✱ p✲❝♦♥✈❡① ✇✐t❤ ❝♦♥st❛♥t C✳ ▲❡t f, g ∈ F ✱ f p ≻ g p ⇒ g E ≤ C f E . Pr♦♦❢✳ ▲❡t f, g ∈ F s✉♣♣♦s❡ t❤❛t f p ≻ g p ✳ ❙✐♥❝❡ . E ✐s ✐♥❝r❡❛s✐♥❣✱ ✇❡ ❝❛♥ ❛•s♦ ❛ss✉♠❡ t❤❛t f p = g p ❜② •❡♠♠❛ ✶✳✹✳✷✳ ❙✐♥❝❡ f ❛♥❞ g ❜❡•♦♥❣ t♦ F ✱ t❤❡r❡ ❡①✐st N, n ∈ N ❛♥❞ ✈❡❝t♦rs a = (a i ) i≤N ❛♥❞ b = (b i ) i≤N ✐♥ (R + ) N s✉❝❤ t❤❛t✿ f = N i=1 a i 1 [(i-1)2 -n ,i2 -n ) ❛♥❞ g = N i=1 b i 1 [(i-1)2 -n ,i2 -n ) . ❚❤❡ ❤②♣♦t❤❡s✐s ♦♥ f ❛♥❞ g ♠❡❛♥s t❤❛t a p ≻ b p ❛♥❞ a p = b p ✳ ❚❤✐s ♠❡❛♥s ❜② t❤❡ ❙❝❤✉r✲❍♦r♥ •❡♠♠❛ t❤❛t b p ✐s ✐♥ t❤❡ ❝♦♥✈❡① ❤✉•• ♦❢ t❤❡ ♣❡r♠✉t❛t✐♦♥s a p σ ♦❢ a p ✇❤❡r❡ ❢♦r σ ∈ S N ✱ ✇❡ ✇r✐t❡ a p σ := (a p σ(i) ) 1≤i≤N ✳ ▲❡t f σ = N i=1 a σ(i) 1 [(i-1)2 -n ,i2 -n ) . ❚❤✐s ♠❡❛♥s t❤❛t t❤❡r❡ ❡①✐st ♥♦♥♥❡❣❛t✐✈❡ ❝♦❡✣❝✐❡♥ts (λ σ ) σ∈S N ❛❞❞✐♥❣ ✉♣ t♦ 1 s✉❝❤ t❤❛t g p = σ∈S N λ σ f p σ = σ∈S N λ 1/p f σ p . ❍❡♥❝❡✱ ❜② p✲❝♦♥✈❡①✐t② ♦❢ E ❛♥❞ ♥♦t✐♥❣ t❤❛t s✐♥❝❡ E ✐s r❡❛rr❛♥❣♠❡♥t ✐♥✈❛r✐❛♥t✱ f σ E = f E ✿ g E ≤ C σ∈S N λ 1/p σ f σ p E 1/p = C f E . ▲❡t ✉s ♥♦✇ st❛t❡ t❤❡ ❦❡② ❛♣♣r♦①✐♠❛t✐♦♥ •❡♠♠❛✳ ▲❡♠♠❛ ✶✳✹✳✹✳ ▲❡t p > 0 ❛♥❞ ε > 0✳ ▲❡t f, g ∈ L 0 (0, α) ❜❡ ♣♦s✐t✐✈❡ ♥♦♥✲ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥s s✉❝❤ t❤❛t f p ≻ g p ✳ ❚❤❡r❡ ❡①✐st s❡q✉❡♥❝❡s (f n ) ❛♥❞ (g n ) ♦❢ ❢✉♥❝t✐♦♥s ✐♥ F s✉❝❤ t❤❛t f n ↑ f ❛✳❡✳ ❛♥❞ g n ↑ g ❛✳❡✳ ❛♥❞ (1 + ε)f p n ≻ g p n ❢♦r ❛•• n ∈ N✳ Pr♦♦❢✳ ❋♦r ❛♥② t ≥ 0 ❛♥❞ n ∈ N✱ ❝♦♥s✐❞❡r t❤❡ ✐♥t❡r✈❛• ♦❢ t❤❡ ❢♦r♠ [i2 -n , (i + 1)2 -n ) ❝♦♥t❛✐♥✐♥❣ t ❛♥❞ ❞❡♥♦t❡ ❜② a n (t) ✐ts r✐❣❤t ❡①tr❡♠✐t②✳ ❉❡✜♥❡ g n (t) = g(a n (t)) ✐❢ t < 2 n 0 ✐❢ t ≥ 2 n ❇② ❝♦♥str✉❝t✐♦♥✱ g n ∈ F ❛♥❞ g n ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣✳ ▼♦r❡♦✈❡r✱ g n (t) ↑ g(t) ❢♦r ❛•• ♣♦✐♥ts ♦❢ ❝♦♥t✐♥✉✐t② ♦❢ g✳ ❙✐♥❝❡ g ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣✱ ✐t ❤❛s ♦♥•② ❝♦✉♥t❛❜•② ♠❛♥② ♣♦✐♥ts ♦❢ ❞✐s❝♦♥t✐♥✉✐t② ❛♥❞ g n ↑ g ❛✳❡✳ ❈♦♥s✐❞❡r ❛•s♦ t❤❡ s❡q✉❡♥❝❡ (f n ) n∈N s✐♠✐•❛r•② ❞❡✜♥❡❞✳ ◆♦✇ ✜① n ❛♥❞ •❡t ✉s ✜♥❞ m n s✉❝❤ t❤❛t (1 + ε)f mn ≻ g n ✳ ❚♦ ❞♦ s♦✱ ♥♦t❡ t❤❛t s✐♥❝❡ ❢♦r ❛•• m✿ F m : t → t 0 f p m ✐s ❝♦♥❝❛✈❡ ❛♥❞ ✐♥❝r❡❛s✐♥❣ ❛♥❞ G n : t → t 0 g p n ✐s ❛✣♥❡ ♦♥ ✐♥t❡r✈❛•s ♦❢ t❤❡ ❢♦r♠ [i2 -n , (i + 1)2 -n ) ❛♥❞ ❝♦♥st❛♥t ❢♦r t •❛r❣❡ ❡♥♦✉❣❤✱ ✐t s✉✣❝❡s t♦ ❝❤❡❝❦ t❤❛t (1 + ε)F m (i2 n ) ≥ G n (i2 n ) ❢♦r ✜♥✐t❡•② ♠❛♥② i ∈ N✳ ❙✐♥❝❡ F m (t) → t 0 f p ❛♥❞ G n (t) ≤ t 0 g p ≤ t 0 f p , ❢♦r m n •❛r❣❡ ❡♥♦✉❣❤✱ ✇❡ ❤❛✈❡ (1 + ε)f mn ≻ g n ✳ ▲❡t i n = max(m n , n)✱ t❤❡ s❡q✉❡♥❝❡s (f in ) ❛♥❞ (g n ) ✈❡r✐❢② t❤❡ r❡q✉✐r❡♠❡♥ts ♦❢ t❤❡ •❡♠♠❛✳ ▲❡♠♠❛ ✶✳✹✳✺✳ ▲❡t p > 0 ❛♥❞ C > 0✳ ▲❡t E ❜❡ ❛ q✉❛s✐✲❇❛♥❛❝❤ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡ ✇✐t❤ t❤❡ ❋❛t♦✉ ♣r♦♣❡rt② s✉❝❤ t❤❛t✿ ∀f, g ∈ F, f p ≻ g p ⇒ g E ≤ C f E . ❚❤❡♥ E ✐s •❡❢t✲p✲♠♦♥♦t♦♥❡✳ Pr♦♦❢✳ ❙✐♥❝❡ E ✐s r❡❛rr❛♥❣❡♠❡♥t ✐♥✈❛r✐❛♥t✱ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t f ❛♥❞ g ❛r❡ ♣♦s✐t✐✈❡ ♥♦♥✲✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥s ❜② ❝♦♥s✐❞❡r✐♥❣ f * ❛♥❞ g * ✳ ▲❡t ε > 0✳ ❚❛❦❡ (f n ) ❛♥❞ (g n ) t✇♦ s❡q✉❡♥❝❡s ❣✐✈❡♥ ❜② •❡♠♠❛ 1.4.4✳ ❋♦r ❛•• n ∈ N✿ g n E ≤ (1 + ε)C f n E ≤ (1 + ε)C f E ✳ ❙♦ ❜② t❤❡ ❋❛t♦✉ ♣r♦♣❡rt②✱ g ∈ E ❛♥❞ g E ≤ (1 + ε)C f E ✳ ❙✐♥❝❡ t❤✐s ✐s tr✉❡ ❢♦r ❛•• ε > 0✱ g E ≤ C f E . ❇② ❝♦♠❜✐♥✐♥❣ t❤❡ r❡s✉•ts ❛❜♦✈❡✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦••♦✇✐♥❣✿ ❈♦r♦••❛r② ✶✳✹✳✻✳ ▲❡t E ❜❡ ❛ q✉❛s✐✲❇❛♥❛❝❤ s②♠♠❡tr✐❝ s❡q✉❡♥❝❡ s♣❛❝❡ ✇✐t❤ t❤❡ ❋❛t♦✉ ♣r♦♣❡rt②✳ ■❢ E ✐s p✲❝♦♥✈❡① t❤❡♥ E ✐s •❡❢t✲p✲♠♦♥♦t♦♥❡✳ ❈♦♥❝❛✈✐t② ✐♠♣•✐❡s r✐❣❤t✲♠♦♥♦t♦♥✐❝✐t② ❚❤❡ s❛♠❡ ❦✐♥❞ ♦❢ ♣r♦♦❢s ❝❛♥ ❜❡ r❡♣r♦❞✉❝❡❞ t♦ ♦❜t❛✐♥ r❡s✉•ts ♦♥ ❝♦♥❝❛✈✐t② ❛♥❞ r✐❣❤t✲♠♦♥♦t♦♥✐❝✐t②✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ •❡♠♠❛✱ s✐♠✐•❛r t♦ •❡♠♠❛ ✶✳✹✳✹✱ ✐s ❣✐✈❡♥ ❜②✿ ▲❡♠♠❛ ✶✳✹✳✼✳ ▲❡t q > 0 ❛♥❞ ε > 0✳ ▲❡t f, g ∈ L 0 (0, ∞) ❜❡ ♣♦s✐t✐✈❡ ♥♦♥✲ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥s s✉❝❤ t❤❛t f q ⊲ g q ✳ ❚❤❡r❡ ❡①✐st s❡q✉❡♥❝❡s (f n ) ❛♥❞ (g n ) ♦❢ ❢✉♥❝t✐♦♥s ✐♥ F s✉❝❤ t❤❛t f n ↑ f ❛✳❡✳✱ g n ↑ g ❛✳❡✳ ❛♥❞ (1 + ε)f q n ⊲ g q n ❢♦r ❛•• n ∈ N✳ ❆♥❞ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣✿ ▲❡♠♠❛ ✶✳✹✳✽✳ ▲❡t E ❜❡ ❛ q✉❛s✐✲❇❛♥❛❝❤ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡✳ • ✐❢ E ✐s q✲❝♦♥❝❛✈❡ ✇✐t❤ ❝♦♥st❛♥t C t❤❡♥ ❢♦r ❛•• f, g ∈ F ✱ f q ⊲ g q ⇒ g E ≤ C f E . • ✐❢ E ❤❛s t❤❡ ❋❛t♦✉ ♣r♦♣❡rt② ❛♥❞ t❤❡r❡ ❡①✐sts C > 0 s✉❝❤ t❤❛t ❢♦r ❛•• f, g ∈ F ✱ f q ⊲ g q ⇒ g E ≤ C f E , t❤❡♥ E ✐s r✐❣❤t✲q✲♠♦♥♦t♦♥❡✳ • ✐❢ E ✐s q✲❝♦♥❝❛✈❡ ❛♥❞ ❤❛s t❤❡ ❋❛t♦✉ ♣r♦♣❡rt②✱ E ✐s r✐❣❤t✲q✲♠♦♥♦t♦♥❡✳ |❡♠❛r❦ ✶✳✹✳✾✳ ❚❤❡ r❡✈❡rs❡ ♦❢ t❤❡ •❡♠♠❛ ❛❜♦✈❡ ❛♥❞ ❝♦r♦••❛r② ✶✳✹✳✻ ❛r❡ ♥♦t tr✉❡✳ ■♥❞❡❡❞✱ ❝♦♥s✐❞❡r ❢♦r ❡①❛♠♣•❡ t❤❡ s♣❛❝❡ L 1,∞ ✳ ■t ✇✐•• ❜❡ ♣r♦✈❡❞ •❛t❡r t❤❛t ✐t ✐s r✐❣❤t✲q✲♠♦♥♦t♦♥❡ ❢♦r ❛•• q > 1 ❜✉t ✐t ✐s ♥♦t q✲❝♦♥❝❛✈❡ ❢♦r ❛♥② q✳ ❈♦♥str✉❝t✐♥❣ s②♠♠❡tr✐❝ s♣❛❝❡s ✇❤✐❝❤ ❛r❡ ♥♦t p✲❝♦♥✈❡① ❢♦r ❛♥② p ✐s ♥♦t ❛♥ ❡❛s② t❛s❦✳ ❲❡ ❛r❡ ✐♥❞❡❜t❡❞ t♦ ❋✳ ❙✉❦♦❝❤❡✈ ❢♦r ✐♥❞✐❝❛t✐♥❣ t❤❡ ❢♦••♦✇✐♥❣ r❡❢❡r❡♥❝❡ t♦ ✉s ✭❬✸✵(✮✳ ■t ✐s ♣♦ss✐❜•❡ t❤❛t s✐♠✐•❛r t❡❝❤♥✐q✉❡s ❝❛♥ ❜❡ ✉s❡❞ t♦ ❝♦♥str✉❝t ❛ •❡❢t✲1✲♠♦♥♦t♦♥❡ s♣❛❝❡ ✇❤✐❝❤ ✐s ♥♦t p✲❝♦♥✈❡① ❢♦r ❛♥② p✳ ✶✳✹✳✷ ❊q✉✐✈❛•❡♥❝❡ ❜❡t✇❡❡♥ ❑✲♠♦♥♦t♦♥✐❝✐t② ❛♥❞ p✲♠♦♥♦t♦♥✐❝✐t② ❋✉♥❝t✐♦♥ s♣❛❝❡s ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♣r♦✈❡ t❤❡ ♠❛✐♥ t❤❡♦r❡♠ ♦❢ t❤✐s ❝❤❛♣t❡r✱ ❛ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ❜❡t✇❡❡♥ L p (0, ∞) ❛♥❞ L q (0, ∞)✳ ❚♦ •✐❣❤t❡♥ t❤❡ ♥♦t❛t✐♦♥s✱ ✇❡ s❡t L r := L r (0, ∞) ❢♦r ❛•• r ∈ (0, ∞]✳ ❚❤❡ ♣r♦♦❢ ✉s❡s ❛ str❛t❡❣② s✐♠✐•❛r ❛s ❬✽(✳ ▲❡t ✉s ✜rst st❛t❡ t❤❡ r❡s✉•t ♣r❡❝✐s❡•②✳ ❚❤❡♦r❡♠ ✶✳✹✳✶✵✳ ▲❡t 0 < p ≤ q < ∞ ❛♥❞ E ❜❡ ❛ q✉❛s✐✲❇❛♥❛❝❤ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡ ♦♥ (0, ∞)✳ ❚❤❡♥ E ✐s •❡❢t✲p✲♠♦♥♦t♦♥❡ ❛♥❞ r✐❣❤t✲q✲♠♦♥♦t♦♥❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ L p ❛♥❞ L q ✳ Pr♦♦❢ ♦❢ t❤❡ r❡✈❡rs❡ ✐♠♣•✐❝❛t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ ❢♦r t❤❡ ♣❛✐r (L p , L q )✳ ❚❤❡♥ ❜② r❡✐t❡r❛t✐♦♥✱ E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❢♦r t❤❡ ♣❛✐r (L p , L ∞ ) ❛♥❞ ❜② ♣r♦♣♦s✐t✐♦♥ ✶✳✸✳✺✱ E ✐s •❡❢t✲p✲♠♦♥♦t♦♥❡✳ ▲❡t f ∈ E ❛♥❞ g ∈ L p + L q s✉❝❤ t❤❛t f q ⊲ g q ✳ ❆ss✉♠❡ t❤❛t f ❛♥❞ g ❛r❡ ♣♦s✐t✐✈❡ ❜② t❛❦✐♥❣ ✐❢ ♥❡❝❡ss❛r② t❤❡✐r ♠♦❞✉•❡s✳ ❲❡ ✇✐•• s❤♦✇ t❤❛t g ∈ E ❛♥❞ g E f E ✳ ❙✐♥❝❡ E ✐s K✲♠♦♥♦t♦♥❡ ❜② ✶✳✸✳✹✱ ✐t ✐s ❡♥♦✉❣❤ t♦ s❤♦✇ t❤❛t ❢♦r ❛•• t > 0✱ K t (g, L p , L q ) K t (f, L p , L q ). ✸✽ ❈❍❆P❚❊| ✶✳ ■◆❚❊|P❖▲❆❚■❖◆ ❙P❆❈❊❙ ❇❊❚❲❊❊◆ L P ✲❙P❆❈❊❙ |❡❝❛•• ❛•s♦ t❤❛t ❜② ✭✶✳✶✮✱ ❢♦r ❛•• h ∈ L p + L q ❛♥❞ t > 0✿ K t (h, p, q) ≈ t α 0 (h * ) p 1/p + t ∞ t α (h * ) q 1/q , ✇❤❡r❡ α = (p -1 -q -1 ) -1 ✳ ▲❡t t > 0✳ ❋✐rst✱ s✐♥❝❡ f q ⊲ g q ✱ t ∞ t α (g * ) q 1/q ≤ t ∞ t α (f * ) q 1/q ≤ K t (f, p, q). ❲❡ ♥♦✇ ❤❛✈❡ t♦ ❡st✐♠❛t❡ t❤❡ s❡❝♦♥❞ t❡r♠ ✐✳❡ t♦ ♣r♦✈❡ t❤❛t✿ t α 0 (g * ) p 1/p K t (f, L p , L q ). ❙✉♣♣♦s❡ ♥♦✇ t❤❛t f ❛♥❞ g ❛r❡ ❜♦✉♥❞❡❞ s♦ t❤❛t f q ❛♥❞ g q ❜❡•♦♥❣ t♦ L 1 ✳ ❙✐♥❝❡ f q ⊲g q ✱ ❜② •❡♠♠❛ ✶✳✹✳✷✱ t❤❡r❡ ❡①✐sts g ′ ≥ g s✉❝❤ t❤❛t g ′ q = f q ❛♥❞ g ′q ≻ f q ✳ ❚❤❡♥ ❜② ❬✻(✱ t❤❡r❡ ❡①✐sts ❛ s✉❜✉♥✐t❛•✱ ♣♦s✐t✐✈❡✱ ✐♥t❡❣r❛• ♣r❡s❡r✈✐♥❣ ♦♣❡r❛t♦r T s✉❝❤ t❤❛t T ((g ′ * ) q ) = (f * ) q ✳ ❍❡♥❝❡ T ((g * ) q ) ≤ (f * ) q ✳ ❉❡✜♥❡ e = 1 (0,t α ) ❛♥❞ ✇r✐t❡✱ t α 0 (g * ) p 1/p = ∞ 0 e(g * ) p 1/p = ∞ 0 T (e(g * ) p ) 1/p . ◆♦t❡ t❤❛t t❤❡r❡ ✐s ❛♥ ❍ö•❞❡r t②♣❡ ✐♥❡q✉❛•✐t② ❢♦r ♣♦s✐t✐✈❡ ✉♥✐t❛• ♦♣❡r❛t♦rs ✐✳❡✱ ❢♦r r, r ′ s✉❝❤ t❤❛t 1 = r -1 + r ′-1 ✱ T (ab) ≤ T (a r ) 1/r T (b r ′ ) 1/r ′ ✳ ❲❡ ❛♣♣•② ✐t t♦ ♦✉r s✐t✉❛t✐♦♥ ✇✐t❤ r = q/p ❛♥❞ r ′ = α/p ❛♥❞ ♦❜t❛✐♥✿ t α 0 (g * ) p 1/p ≤ ∞ 0 T (e) p/α T ((g * ) q ) p/q 1/p t α 0 (f * ) p T (e) p/α 1/p + ∞ t α (f * ) p T (e) p/α 1/p ❚♦ ❡st✐♠❛t❡ t❤❡ •❡❢t s✉♠♠❛♥❞✱ ✇❡ ✉s❡ t❤❡ ❢❛❝t t❤❛t T e ∞ ≤ 1 ❛♥❞ ❢♦r t❤❡ r✐❣❤t s✉♠♠❛♥❞✱ ✇❡ ❛♣♣•② t❤❡ ✉s✉❛• ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t②✳ ≤ t α 0 (f * ) p 1/p + ∞ t α (f * ) q 1/q ∞ t α T (1 (0,t α ) ) 1/α K t (f, L p , L q ) ✇❤❡r❡ ✇❡ ✉s❡❞ t❤❛t ∞ t α T (1 (0,t α ) ) 1/α ≤ ∞ 0 T (1 (0,t α ) ) 1/α = t✳ ❚♦ ❝♦♥❝•✉❞❡ ❢♦r ✉♥❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s✱ ✐t s✉✣❝❡s t♦ ❛♣♣r♦①✐♠❛t❡ g * ❛♥❞ f * ✳ ❆ ✇❛② t♦ ❞♦ ✐t ✐s t♦ ❛♣♣•② t❤❡ ♣r❡✈✐♦✉s ✐♥❡q✉❛•✐t② t♦ g * (. -ε) ❛♥❞ f * (. -ε) ❛♥❞ •❡t ε ❣♦ t♦ 0✳ ✶✳✹✳ P ✲▼❖◆❖❚❖◆■❈■❚( ✸✾ Pr♦✈✐♥❣ t❤❡ ❞✐r❡❝t ✐♠♣•✐❝❛t✐♦♥ r❡q✉✐r❡s ♠♦r❡ ✇♦r❦✳ ❚❤❡ ✜rst •❡♠♠❛ ✐s ❛♥ ❡❛s② ❝❤❡❝❦✳ ❲❡ ♥❡❡❞ t♦ ✈❡r✐❢② t❤❛t E ✐s ❛♥ ✐♥t❡r♠❡❞✐❛t❡ s♣❛❝❡ ❢♦r t❤❡ ❝♦✉♣•❡ (L p , L q )✳ ▲❡♠♠❛ ✶✳✹✳✶✶✳ ▲❡t q ≥ p > 0 ❛♥❞ C > 0✳ ▲❡t E ❜❡ ❛ q✉❛s✐✲❇❛♥❛❝❤ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡✳ ❙✉♣♣♦s❡ t❤❛t E ✐s •❡❢t✲p✲♠♦♥♦t♦♥❡ ❛♥❞ r✐❣❤t✲q✲♠♦♥♦t♦♥❡✳ ❚❤❡♥ L p ∩ L q ⊂ E ⊂ L p + L q . Pr♦♦❢✳ ▲❡t ✉s ♣r♦✈❡ t❤❡ ✜rst ✐♥❝•✉s✐♦♥✳ ❋✐rst ♥♦t❡ t❤❛t s✐♥❝❡ E ✐s ❛ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡✱ ✐t ❝♦♥t❛✐♥s ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥s ♦❢ s❡ts ♦❢ ✜♥✐t❡ ♠❡❛s✉r❡ ✭❬✶✶(✮✳ ▲❡t f ∈ L p ∩ L q ✳ ❉❡❝♦♠♣♦s❡ f ✐♥t♦ f 1 = f 1 |f |>1 ❛♥❞ f 2 = f 1 |f |≤1 ✳ ▲❡t h 1 = 1 0, f1 q q ✳ ❙✐♥❝❡ f 1 ∈ L q ✱ h 1 ∈ E ❛♥❞ ✐t ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ ❝❤❡❝❦ t❤❛t h q 1 ⊲ f q 1 ✳ ■♥❞❡❡❞✱ ❢♦r ❛•• t ∈ [0, f 1 q q )✱ ∞ t (f * 1 ) q = f 1 q q - t 0 f * 1 ≥ f 1 q q - t 0 1 = ∞ t (h * 1 ) q . ❆♥❞ ❢♦r t ≥ f 1 q q ✱ ∞ t (h * 1 ) q = 0 ≤ ∞ t (f * 1 ) q ✳ ❙♦ h q 1 ⊲f q 1 ✱ ❤❡♥❝❡ f 1 ∈ E✳ ❙✐♠✐•❛r•② •❡t h 2 = 1 0, f2 p p ✱ h 2 ✐s ✐♥ E ❛♥❞ h p 2 ≻ f p 2 s♦ f 2 ∈ E✳ ❍❡♥❝❡ f ∈ E✳ ▲❡t ✉s ♥♦✇ ♣r♦✈❡ t❤❡ s❡❝♦♥❞ ✐♥❝•✉s✐♦♥✳ ▲❡t f / ∈ L p + L q ✳ ❉❡❝♦♠♣♦s❡ f ✐♥t♦ f 1 ❛♥❞ f 2 ❛s ❛❜♦✈❡✳ ❇② ❛ss✉♠♣t✐♦♥✱ f 1 / ∈ L p + L q ♦r f 2 / ∈ L p + L q ✳ ❝❛s❡ ✶✿ f 1 / ∈ L p + L q ✳ ❚❤❡♥ f 1 ✐s ♥♦t ✐♥ L p ✱ t❤✐s ♠❡❛♥s t❤❛t f p 1 ≻ a1 [0,1] ❢♦r ❛•• a > 0✱ ❤❡♥❝❡ f 1 E = ∞✱ f 1 / ∈ E✳ ❝❛s❡ ✷✿ f 2 / ∈ L p + L q ✳ ❙✐♠✐•❛r•②✱ f 2 / ∈ L q s♦ f q 2 ⊲ a1 [0,1] ❢♦r ❛•• a > 0 s♦ f 2 / ∈ E✳ ❚❤❡ ❢♦••♦✇✐♥❣ t✇♦ •❡♠♠❛s ❝♦♥st✐t✉t❡ t❤❡ ❤❡❛rt ♦❢ t❤❡ ♣r♦♦❢✳ ❚❤❡② ✇✐•• ❜❡ ✉s❡❞ t♦ ♣r♦✈❡ t❤❛t •❡❢t✲p✲♠♦♥♦t♦♥✐❝✐t② ❛♥❞ r✐❣❤t✲q✲♠♦♥♦t♦♥✐❝✐t② ✐♠♣•② K✲ ♠♦♥♦t♦♥✐❝✐t② ❢♦r t❤❡ ♣❛✐r (L p , L q )✳ ❋♦r A ❛ t♦♣♦•♦❣✐❝❛• s♣❛❝❡ ❛♥❞ x ∈ A✱ ✇❡ ✇✐•• ❞❡♥♦t❡ ❜② A(x) t❤❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ A ❝♦♥t❛✐♥✐♥❣ x✳ ▲❡♠♠❛ ✶✳✹✳✶✷✳ ▲❡t A ❛♥❞ B ❜❡ t✇♦ ♦♣❡♥ s✉❜s❡ts ♦❢ (0, ∞) s✉❝❤ t❤❛t A ∪ B = (0, ∞)✳ ❚❤❡♥ t❤❡r❡ ❡①✐st ✐♥t❡r✈❛•s I, J ⊂ Z ❛♥❞ s❡q✉❡♥❝❡s ♦❢ ♦♣❡♥ ✐♥t❡r✈❛•s (A i ) i∈I ❛♥❞ (B j ) j∈J ✐♥ (0, ∞) s✉❝❤ t❤❛t✿ • ❢♦r ❛•• i ∈ I✱ A i = (a i , a ′ i ) ✐s ❛ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ A ❛♥❞ ❢♦r ❛•• j ∈ J✱ B j = (b j , b ′ j ) ✐s ❛ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ B✱ • ❢♦r ❛•• n ∈ Z✱ ✐❢ n ∈ I ∩ J t❤❡♥ a n ∈ B n ❛♥❞ ✐❢ n ∈ I ❛♥❞ n + 1 ∈ J t❤❡♥ b n+1 ∈ A n ✱ • ✐❢ n ∈ I ❛♥❞ n -1 ∈ I✱ t❤❡♥ n ∈ J ❛♥❞ a n-1 < b n < a n ✳ ❙✐♠✐•❛r•②✱ ✐❢ n ∈ J ❛♥❞ n + 1 ∈ J t❤❡♥ n ∈ A ❛♥❞ b n < a n < b n+1 ✳ • i∈I A i ∪ j∈J B j = (0, ∞)✳ ✹✵ ❈❍❆P❚❊| ✶✳ ■◆❚❊|P❖▲❆❚■❖◆ ❙P❆❈❊❙ ❇❊❚❲❊❊◆ L P ✲❙P❆❈❊❙ ❚❤❡s❡ ❝♦♥❞✐t✐♦♥s✱ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ✐♥❞✐❝❡s✱ ❝❛♥ ❜❡ ❞✐✣❝✉•t t♦ ♣✐❝t✉r❡ ❛t ✜rst✳ ❲❤❛t ✇❡ ✇❛♥t t♦ ♣r♦✈❡ ✐s ❡ss❡♥t✐❛••② t❤❛t t❤❡ s✐t✉❛t✐♦♥ ✐s s✐♠✐•❛r t♦ t❤❡ ✜❣✉r❡ ❜❡•♦✇✳ x 0 • b 0 B 0 b ′ 0 a ′ -1 A -1 a -1 a 0 A 0 a ′ 0 b 1 B 1 b ′ -1 B -1 ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ Pr♦♦❢✳ ▲❡t ✉s ❝♦♥str✉❝t (A n )✱ (B n )✱ I ❛♥❞ J ❜② ✐♥❞✉❝t✐♦♥✳ ■❢ A = (0, ∞)✱ t❤❡r❡ ✐s ♥♦t❤✐♥❣ t♦ ❞♦ s♦ s✉♣♣♦s❡ t❤❛t t❤❡r❡ ❡①✐sts x 0 ∈ B\A ❛♥❞ ❞❡✜♥❡ B 0 = B(x 0 )✳ ❚❤❡r❡ ✐s ♥♦✇ ❡ss❡♥t✐❛••② ♦♥❡ ♣♦ss✐❜•❡ ❝♦♥str✉❝t✐♦♥ s❛st✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s ❛❜♦✈❡✳ ▲❡t ✉s ❞❡t❛✐• t❤❡ ✜rst st❡♣✿ ✐❢ b 0 = 0✱ t❤❡♥ b 0 ∈ A ❛♥❞ t❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ ✐♠♣•✐❡s t❤❛t ✐t ❜❡•♦♥❣s t♦ A -1 ✳ ❚❤❡♥✱ ❜② t❤❡ ✜rst ❝♦♥❞✐t✐♦♥✱ ✇❡ ❤❛✈❡ t♦ ❞❡✜♥❡✱ A -1 = A(b 0 )✳ ■❢ a -1 = 0✱ t❤❡ ❡①t❡♥s✐♦♥ t♦ t❤❡ •❡❢t ❝♦♥t✐♥✉❡s ✇✐t❤ B -1 = B(a -1 )✳✳✳ ❚❤✐s •❡❛❞s t♦ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦❝❡❞✉r❡✱ ✐❢ a n = 0 ♦r b n = 0 ✐s t❤❡ ♣♦✐♥t t❤❡ ♠♦r❡ t♦ t❤❡ •❡❢t t❤❛t ❤❛s ❛•r❡❛❞② ❜❡❡♥ ❞❡✜♥❡❞✱ ❡①t❡♥❞ t♦ t❤❡ •❡❢t ❜② ❞❡✜♥✐♥❣ A n-1 = A(b n ) ❛♥❞ B n = B(a n )✳ ❚❤❡ ♣r♦❝❡❞✉r❡ st♦♣s ✐❢ ❛t s♦♠❡ ♣♦✐♥t a n = 0 ♦r b n = 0✳ ❊•s❡✱ ✇❡ ❛•✇❛②s ❤❛✈❡ a n / ∈ A s♦ a n ∈ B ✭s✐♠✐•❛r•②✱ b n ∈ A✮ ❛♥❞ t❤❡ ♣r♦❝❡❞✉r❡ ✐s ✇❡••✲❞❡✜♥❡❞✳ ❚❤❡ ❡①t❡♥s✐♦♥ t♦ t❤❡ r✐❣❤t ✐s s✐♠✐•❛r✳ ■❢ b ′ n = ∞ ♦r a ′ n-1 = ∞ ✐s t❤❡ ♣♦✐♥t t❤❡ ♠♦r❡ t♦ t❤❡ r✐❣❤t✱ ❞❡✜♥❡ A n = A(b ′ n ) ❛♥❞ B n = B(a ′ n-1 )✳ ❚❤✐s t✐♠❡✱ t❤❡ ♣r♦❝❡❞✉r❡ st♦♣s ✐❢ a ′ n = ∞ ♦r b ′ n = ∞ ❛♥❞ ✐t ✐s st✐•• ✇❡•• ❞❡✜♥❡❞✳ ▲❡t ✉s ❝❤❡❝❦ t❤❛t t❤❡ ♣r♦♣❡rt✐❡s ❛r❡ ✈❡r✐✜❡❞✳ ❈♦♥❞✐t✐♦♥ ✶✳ ■t ✐s tr✉❡ ❜② ❞❡✜♥✐t✐♦♥✳ ❈♦♥❞✐t✐♦♥ ✷✳ ❲❡ ♦♥•② ❣✐✈❡ ❞❡t❛✐•s ❢♦r t❤❡ ✜rst ♣r♦♣❡rt②✳ ■❢ n ∈ I ∩ J t❤✐s ♠❡❛♥s t❤❛t a n = 0✳ ■❢ n < 0✱ ✇❡ ❤❛✈❡ B n = B(a n ) s♦ a n ∈ B n ✳ ■❢ n = 0✱ A 0 = A(b ′ 0 )✳ ❙♦ a 0 < b ′ 0 ❛♥❞ ❜② ❛ss✉♠♣t✐♦♥ ♦♥ x 0 ✱ a 0 > x 0 > b 0 s♦ a 0 ∈ B 0 ✳ ❋✐♥❛••②✱ ✐❢ n > 0 t❤❡♥ B n = B(a ′ n-1 ) ❛♥❞ A n = A(b ′ n ) s♦ b n < a ′ n-1 < a n < b ′ n ✱ a n ∈ B n ✳ ❈♦♥❞✐t✐♦♥ ✸✳ ❙✉♣♣♦s❡ t❤❛t n ∈ I ❛♥❞ n -1 ∈ I✳ ❚❤✐s ✐♠♣•✐❡s t❤❛t a n = 0 ❛♥❞ a ′ n-1 = ∞✳ ❙♦ a n ∈ B s♦✱ ✉s✐♥❣ t❤❡ t✇♦ ♣r❡✈✐♦✉s ♣r♦♣❡rt✐❡s✱ B n = B(a n ) s♦ b n < a n ✳ ❆♥❞ A n-1 = A(b n ) s♦ a n-1 < b n ✳ ❚❤❡ s❡❝♦♥❞ ♣r♦♣❡rt② ✐s ♣r♦✈❡❞ ❜② ❛ s✐♠✐•❛r ❛r❣✉♠❡♥t✳ ❈♦♥❞✐t✐♦♥ ✹✳ ❲❡ ♦♥•② ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ ✇❤❡r❡ I = J = Z s✐♥❝❡ t❤❡ ♦t❤❡r ❝❛s❡s ❛r❡ ❡❛s✐❡r✳ ■t s✉✣❝❡s t♦ ♣r♦✈❡ t❤❛t t❤❡ a ′ i → ∞ ✇❤❡♥ i → ∞ ❛♥❞ a i → 0 ✇❤❡♥ i → -∞ s✐♥❝❡ n∈N A n ∪ B n ✐s ❝♦♥♥❡❝t❡❞✳ ❆ss✉♠❡ ❜② ❝♦♥tr❛❞✐❝t✐♦♥ t❤❛t t❤❡ (a ′ n ) n≥0 ❛❝❝✉♠✉•❛t❡s ❛t s♦♠❡ ♣♦✐♥t x✳ ❚❤❡♥ t❤❡ (b ′ n ) ❛•s♦ ❛❝❝✉♠✉•❛t❡s ❛t x✳ ❇✉t x ∈ A∪B ✇❤✐❝❤ ❛r❡ ♦♣❡♥ s❡t s♦ ❢♦r n •❛r❣❡ ❡♥♦✉❣❤ ✇❡ ❡✐t❤❡r ❤❛✈❡ a ′ n ∈ A(x) ♦r b ′ n ∈ B(x) ✇❤✐❝❤ ✐s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❙✐♠✐•❛r•②✱ ✇❡ ❣❡t ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ✐❢ ✇❡ s✉♣♣♦s❡ t❤❛t t❤❡ s❡q✉❡♥❝❡ (a n ) n≤0 ❛❝❝✉♠✉•❛t❡s ❛t ❛ ♣♦✐♥t x > 0✳ ❲❡ ✉s❡ t❤❡ ♣r❡✈✐♦✉s ❝♦♥str✉❝t✐♦♥ t♦ ♦❜t❛✐♥ t❤❡ ❢♦••♦✇✐♥❣ ❦❡② •❡♠♠❛✳ ✶✳✹✳ P ✲▼❖◆❖❚❖◆■❈■❚( ✹✶ ▲❡♠♠❛ ✶✳✹✳✶✸✳ ▲❡t 0 < p ≤ q < ∞✱ α > 0 ❛♥❞ f, g ∈ (L p + L q ) + ❜❡ ♥♦♥✲ ✐♥❝r❡❛s✐♥❣ r✐❣❤t✲❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s s✉❝❤ t❤❛t ❢♦r ❛•• t > 0✿ t α 0 f p 1/p + t ∞ t α f q 1/q > t α 0 g p 1/p + t ∞ t α g q 1/q , t❤❡♥ t❤❡r❡ ❡①✐st ♥♦♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥s h ❛♥❞ l s✉❝❤ t❤❛t h + l = g✱ h ✐s ♥♦♥✲ ✐♥❝r❡❛s✐♥❣✱ f p ≻ h p ❛♥❞ f q ⊲ l q ✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ❢♦r ❛•• t > 0✱ ∞ t f q ≥ ∞ t l q . Pr♦♦❢✳ ▲❡t A := {t ∈ R + : t 0 f p > t 0 g p } ❛♥❞ B := {t ∈ R + : ∞ t f q > ∞ t g q }✳ A ❛♥❞ B ❛r❡ ♦♣❡♥ s❡ts ❛♥❞ A ∪ B = (0, ∞)✳ ▲❡t (A n ) n∈Z ❛♥❞ (B n ) n∈Z ❜❡ ♦❜t❛✐♥❡❞ ❜② t❤❡ •❡♠♠❛ ❛❜♦✈❡✳ ❲❡ ♦♥•② ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ ♦❢ ✐♥✜♥✐t❡ ❢❛♠✐•✐❡s ♦❢ ✐♥t❡r✈❛•s ✐♥ ❜♦t❤ ❞✐r❡❝t✐♦♥s✱ t❤❡ ♦t❤❡r ❝❛s❡s ❛r❡ s✐♠✐•❛r✳ ❉❡✜♥❡ h ❛♥❞ l ❛s ❢♦••♦✇s✳ ❋♦r ❛•• n ∈ Z✱ h(t) = g(a n ) ✐❢ t ∈ [b n , a n ) g(t) ✐❢ t ∈ [a n , b n+1 ) ▲❡t l = g -h✳ g ❛♥❞ l ❛r❡ ❝•❡❛r•② ♥♦♥❡❣❛t✐✈❡✳ ◆♦t❡ t❤❛t ❢♦r ❛•• n ∈ Z✱ f (a n ) p ≥ g(a n ) p ✳ ■♥❞❡❡❞✱ s✐♥❝❡ a n ✐s ❛♥ ❡①tr❡♠✐t② ♦❢ ❛ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ A✱ an 0 f p = an 0 g p ❛♥❞ ❢♦r ε s♠❛•• ❡♥♦✉❣❤ s♦ t❤❛t a n + ε < a ′ n ✱ an+ε 0 f p > an+ε 0 g p ✳ ❍❡♥❝❡✱ an+ε an f p > an+ε an g p ✳ ❙✐♥❝❡ f ❛♥❞ g ❛r❡ r✐❣❤t✲❝♦♥t✐♥✉♦✉s t❤✐s ✐♠♣•✐❡s t❤❛t f (a n ) p ≥ g(a n ) p ✳ ▲❡t ✉s ♥♦✇ ❝❤❡❝❦ t❤❛t t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✈❡r✐✜❡s ✇❤❛t ✇❡ ❝•❛✐♠❡❞✳ ❋♦r ❛•• n ∈ N✱ ❢♦r ❛•• t ∈ [b n , a n )✿ t 0 h p = bn 0 h p + t bn h p ≤ bn 0 g p + t bn g(a n ) ≤ bn 0 f p + t bn f (a n ) p ≤ t 0 f p , ❛♥❞ s✐♥❝❡ a n ∈ B n ✱ t ∈ B ❛♥❞✿ ∞ t l q ≤ ∞ t g q ≤ ∞ t f q . ❋♦r ❛•• t ∈ [a n , b n+1 )✱ s✐♥❝❡ b n+1 ∈ A n ✱ t ∈ A ✇❤✐❝❤ ✐♠♣•✐❡s t❤❛t✿ t 0 h p ≤ t 0 g p ≤ t 0 f p . ✹✷ ❈❍❆P❚❊| ✶✳ ■◆❚❊|P❖▲❆❚■❖◆ ❙P❆❈❊❙ ❇❊❚❲❊❊◆ L P ✲❙P❆❈❊❙ ❛♥❞ ∞ t l q = ∞ bn+1 l q ≤ ∞ bn+1 g q ≤ ∞ bn+1 f q . ▼♦r❡♦✈❡r✱ ♥♦t❡ t❤❛t h ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣✱ s♦ ❢♦r ❛•• t ≥ 0✿ t 0 (h * ) p = t 0 h p ≤ t 0 f p = t 0 (f * ) p . ❆♥❞ s✐♥❝❡ l ✐s ♣♦s✐t✐✈❡✱ ❢♦r ❛•• t ≥ 0✿ ∞ t (l * ) q ≤ ∞ t l q ≤ ∞ t f q = ∞ t (f * ) q . ❲❡ ❛r❡ ♥♦✇ r❡❛❞② t♦ ✜♥✐s❤ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠✳ ❊♥❞ ♦❢ t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✶✳✹✳✶✵✳ ❇② ♣r♦♣♦s✐t✐♦♥ ✶✳✷✳✺✱ ✐t s✉✣❝❡s t♦ ♣r♦✈❡ t❤❛t E ✐s K✲♠♦♥♦t♦♥❡✳ ❲❡ ❛•r❡❛❞② ❦♥♦✇ ❢r♦♠ •❡♠♠❛ ✶✳✹✳✶✶ t❤❛t L p ∩ L q ⊂ E ⊂ L p + L q ✳ ❆ss✉♠❡ t❤❛t f = 0✳ ❙✉♣♣♦s❡ t❤❛t ❢♦r ❛•• t > 0✱ K t (f, L p , L q ) ≥ K t (g, L p , L q )✳ ❇② ✭✶✳✶✮✱ t❤✐s ♠❡❛♥s t❤❛t t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c > 0 ✐♥❞❡♣❡♥✲ ❞❡♥t ♦❢ f ❛♥❞ g s✉❝❤ t❤❛t ❢♦r ❛•• t > 0✿ t α 0 (cf * ) p 1/p + t ∞ t α (cf * ) q 1/q > t α 0 (g * ) p 1/p + t ∞ t α (g * ) q 1/q , ❙♦ ❜② •❡♠♠❛ ✶✳✹✳✶✸ ❛♣♣•✐❡❞ t♦ f * ❛♥❞ g * ✱ t❤❡r❡ ❡①✐sts t✇♦ ❢✉♥❝t✐♦♥s h ❛♥❞ l s✉❝❤ t❤❛t h + l = g * ✱ (cf ) p ≻ h p ❛♥❞ (cf ) q ⊲ l q ✳ ❍❡♥❝❡✱ ❜② t❤❡ ❛ss✉♠♣t✐♦♥s ♦♥ E✱ h, l ∈ E s♦ g ∈ E ❛♥❞ g E = g * E h E + l E ≤ 2cK f E . |❡♠❛r❦ ✶✳✹✳✶✹✳ ❚❤❡ r❡s✉•t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞✱ ✇✐t❤ t❤❡ s❛♠❡ ♣r♦♦❢✱ t♦ s②♠♠❡t✲ r✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡s ♦♥ t❤❡ ✐♥t❡r✈❛• (0, α)✱ α ∈ (0, ∞)✳ |❡♠❛r❦ ✶✳✹✳✶✺✳ ■t ✐s ❦♥♦✇♥ t❤❛t ❛ q✲❝♦♥❝❛✈❡ s♣❛❝❡ ✐s ❛✉t♦♠❛t✐❝❛••② p✲❝♦♥✈❡① ❢♦r s♦♠❡ p > 0✱ ✐t ✐s ❢♦r ❡①❛♠♣•❡ ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡♦r❡♠ ✹✳✶ ✐♥ ❬✷✽(✳ ❈♦♥•❡❝t✉r❡ ✶✳✶✳✶ ❢♦r ❢✉♥❝t✐♦♥ s♣❛❝❡s✱ ✐s ♥♦✇ r❡❞✉❝❡❞ t♦ ♣r♦✈✐♥❣ ❛ s✐♠✐•❛r ❢❛❝t ❢♦r r✐❣❤t✲q✲ ♠♦♥♦t♦♥✐❝✐t② ❛♥❞ •❡❢t✲p✲♠♦♥♦t♦♥✐❝✐t②✳ |❡♠❛r❦ ✶✳✹✳✶✻✳ ❆s ❛ ❝♦r♦••❛r② ♦❢ t❤❡♦r❡♠ ✶✳✹✳✶✵ ❛♥❞ ❝♦r♦••❛r② ✶✳✹✳✻✱ ✇❡ ♦❜t❛✐♥ t❤❡♦r❡♠ ✶✳✶✳✷ ♠❡♥t✐♦♥♥❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥✳ ✶✳✹✳ P ✲▼❖◆❖❚❖◆■❈■❚( ✹✸ ❙❡q✉❡♥❝❡ s♣❛❝❡s ❲❡ ✇✐•• ♥♦✇ ♣r♦✈❡ t❤❡ ♠❛✐♥ t❤❡♦r❡♠ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ s②♠♠❡tr✐❝ s❡q✉❡♥❝❡ s♣❛❝❡s r❛t❤❡r t❤❛♥ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡s✳ ■♥ t❤✐s s❡tt✐♥❣✱ ✐t ✐s ❡❛s② t♦ s❡❡ t❤❛t t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛r❣✉♠❡♥ts ♠❛❞❡ ❛❜♦✈❡ st✐•• ❤♦•❞✳ ❲❡ ✇✐•• ❛•s♦ ✉s❡ t❤❡ ❢❛❝t t❤❛t ℓ ∞ ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥t♦ L ∞ ❜②✿ u → ∞ i=1 u i 1 [i-1,i) . ❍❡♥❝❡✱ ✇❡ ✇✐•• ✐❞❡♥t✐❢② ℓ ∞ ✇✐t❤ t❤❡ s✉❜❛•❣❡❜r❛ ♦❢ L ∞ ♦❢ ❢✉♥❝t✐♦♥s ❛✳❡✳ ❝♦♥st❛♥t ♦♥ ✐♥t❡r✈❛•s ♦❢ t❤❡ ❢♦r♠ [i, i+1)✱ i ∈ N✳ ❉❡♥♦t❡ ❜② E t❤❡ ❝♦♥❞✐t✐♦♥❛• ❡①♣❡❝t❛t✐♦♥ ❢r♦♠ L ∞ t♦ ℓ ∞ ✳ ❚❤❡ ♥♦t✐♦♥s ♦❢ •❡❢t ❛♥❞ r✐❣❤t ♠❛•♦r✐③❛t✐♦♥ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ ✇✐t❤♦✉t ♠♦❞✐✜❝❛t✐♦♥s t♦ s❡q✉❡♥❝❡s ❛♥❞ ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ✉s✉❛• ♥♦t✐♦♥✳ ▲❡t u ❛♥❞ v ✐♥ ℓ ∞ ✿ u ≻ v ⇔ ∀n ∈ N, n i=0 u * i ≥ n i=0 v * i ❛♥❞ u ⊲ v ⇔ ∀n ∈ N, ∞ i=n u * i ≥ ∞ i=n v * i . ▲❡t ✉s ♥♦✇ st❛t❡ t❤❡ r❡s✉•t ♦❢ t❤✐s s✉❜s❡❝t✐♦♥✿ ❚❤❡♦r❡♠ ✶✳✹✳✶✼✳ ▲❡t 0 < p ≤ q < ∞ ❛♥❞ E ❜❡ ❛ q✉❛s✐✲❇❛♥❛❝❤ s②♠♠❡tr✐❝ s❡q✉❡♥❝❡ s♣❛❝❡✳ ■❢ E ✐s •❡❢t✲p✲♠♦♥♦t♦♥❡ ❛♥❞ r✐❣❤t✲q✲♠♦♥♦t♦♥❡ t❤❡♥ E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ ℓ p ❛♥❞ ℓ q ✳ Pr♦♦❢✳ ❇② ♣r♦♣♦s✐t✐♦♥ ✶✳✷✳✺✱ ✐t s✉✣❝❡s t♦ ❝❤❡❝❦ t❤❛t E ✐s K✲♠♦♥♦t♦♥❡✳ ❙✐♠✐•❛r•② t♦ t❤❡ ❝❛s❡ ♦❢ ❢✉♥❝t✐♦♥ s♣❛❝❡s✱ t❤❡ ❤②♣♦t❤❡s✐s ♦❢ t❤❡ t❤❡♦r❡♠ ✐♠♣•② t❤❛t ℓ p = ℓ p ∩ ℓ q ⊂ E ⊂ ℓ p + ℓ q = ℓ q ❛♥❞ t❤❛t ❢♦r ❛♥② u ∈ E ❛♥❞ v ∈ ℓ q ✿ u q ⊲ v q ⇒ v ∈ E, v E ≤ C u E ❛♥❞ u p ≻ v p ⇒ v ∈ E, v E ≤ C u E . ▲❡t u ∈ E ❛♥❞ v ∈ ℓ q ✱ u = 0✳ ❆ss✉♠❡ t❤❛t ❢♦r ❛•• t > 0, K t (u, ℓ p , ℓ q ) ≤ K t (v, ℓ p , ℓ q )✳ ❚❤❡ ❢♦r♠✉•❛ ❢♦r t❤❡ K✲❢✉♥❝t✐♦♥❛• ✉s❡❞ ♣r❡✈✐♦✉s•② st✐•• ❤♦•❞s ✐♥ t❤✐s ❝♦♥t❡①t✳ ■♥❞❡❡❞✱ ❢♦r ❛♥② s❡q✉❡♥❝❡ w ∈ ℓ q ❛♥❞ t > 0✿ K t (w, p, q) ≈ t α 0 (w * ) p 1/p + t ∞ t α (w * ) q 1/q . ❙♦ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c ✐♥❞❡♣❡♥❞❡♥t ♦❢ u ❛♥❞ v s♦ t❤❛t ❢♦r ❛•• t > 0✿ t α 0 (cu * ) p 1/p + t ∞ t α (cu * ) q 1/q > t α 0 (v * ) p 1/p + t ∞ t α (v * ) q 1/q , ❇② •❡♠♠❛ ✶✳✹✳✶✸✱ t❤❡r❡ ❡①✐sts h ❛♥❞ l ♥♦♥✲♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥s s✉❝❤ t❤❛t✱ h + l = v * ✱ h ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣✱ (cu) p ≻ h p ❛♥❞ (cu) q ⊲ l q ✳ ▲❡t h = (E(h p )) 1/p ❛♥❞ l = E(l q ) 1/q ✳ ❲❡ ✇✐•• st❛rt ❜② s❤♦✇✐♥❣ t❤❛t (cu) p ≻ h p ❛♥❞ (cu) q ⊲ g q ✳ ❚❤❡ ❢✉♥❝t✐♦♥s✿ t → t 0 (cu * ) p ❛♥❞ t → t 0 h p ✹✹ ❈❍❆P❚❊| ✶✳ ■◆❚❊|P❖▲❆❚■❖◆ ❙P❆❈❊❙ ❇❊❚❲❊❊◆ L P ✲❙P❆❈❊❙ ❛r❡ ❛✣♥❡ ♦♥ ✐♥t❡r✈❛•s ♦❢ t❤❡ ❢♦r♠ [i, i + 1]✱ i ∈ N✳ ❙♦ ✐t s✉✣❝❡s t♦ ❝❤❡❝❦ t❤❡ ✐♥❡q✉❛•✐t② ❛t ✐♥t❡❣❡rs✳ ▲❡t n ∈ N✱ n 0 (cu * ) p ≥ n 0 h p = n 0 h p . ❚❤❡ ♦t❤❡r ✐♥❡q✉❛•✐t② ✐s ♣r♦✈❡♥ ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r✳ |❡❝❛•• t❤❛t h ❛♥❞ l ❝❛♥ ❜❡ s❡❡♥ ❛s s❡q✉❡♥❝❡s✳ ◆♦t❡ t❤❛t a p := max(1, 2 p-1 ) ✐s t❤❡ ♦♣t✐♠❛• ❝♦♥st❛♥t s✉❝❤ t❤❛t ❢♦r ❛•• a, b ∈ R + ✱ (a + b) p ≤ a p (a p + b p )✳ ❲❡ ❤❛✈❡✿ v * = (E((v * ) p )) 1/p ≤ a p E(h p +l p ) 1/p ≤ a p a 1/p E(h p ) 1/p + E(l p ) 1/p ≤ a p a 1/p (h+l). ❆♥❞ ❜② ❤②♣♦t❤❡s✐s ♦♥ E✱ h ∈ E ❛♥❞ l ∈ E ✇✐t❤ h E ≤ cK u E ❛♥❞ l E ≤ cK u E ✳ ❙♦ v ∈ E ❛♥❞ v E ≤ u E ✇✐t❤ ❛♥ ✐♠♣•✐❝✐t ❝♦♥st❛♥t ❞❡♣❡♥❞✐♥❣ ♦♥ E✱ p ❛♥❞ q✳ |❡♠❛r❦ ✶✳✹✳✶✽✳ ◆♦t❡ t❤❛t K✲♠♦♥♦t♦♥✐❝✐t② ✐s ♥♦t ❦♥♦✇♥ ✐♥ ❣❡♥❡r❛• ❢♦r ❝♦✉♣•❡s ♦❢ ℓ p ✲s♣❛❝❡s✳ ❚❤✐s ✐s ✇❤② ✇❡ ❝❛♥♥♦t s❤♦✇ t❤❡ r❡✈❡rs❡ ✐♠♣•✐❝❛t✐♦♥ ✐♥ t❤❡♦r❡♠ ✶✳✹✳✶✼✳ ❍♦✇❡✈❡r✱ ♥♦t❡ t❤❛t ❢♦r 1 ≤ p ≤ q ≤ ∞✱ t❤❡ ❝♦✉♣•❡ (ℓ p , ℓ q ) ✐s ❛ ❈❛•❞❡ró♥ ❝♦✉♣•❡ ✭✶✳✸✳✹✮ ❛♥❞ ✇❡ ❝❛♥ r❡❝♦✈❡r t❤❡ ❢♦••♦✇✐♥❣✿ ❈♦r♦••❛r② ✶✳✹✳✶✾✳ ▲❡t 1 ≤ p ≤ q < ∞ ❛♥❞ E ❜❡ ❛ q✉❛s✐✲❇❛♥❛❝❤ s②♠♠❡tr✐❝ s❡q✉❡♥❝❡ s♣❛❝❡✳ ❚❤❡♥ E ✐s •❡❢t✲p✲♠♦♥♦t♦♥❡ ❛♥❞ r✐❣❤t✲q✲♠♦♥♦t♦♥❡ ✐❢ ❛♥❞ ♦♥•② ✐❢ E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ ℓ p ❛♥❞ ℓ q ✳ ❚❤✐s ✐s ❝•♦s❡ t♦ t❤❡ ♠❛✐♥ r❡s✉•t ♣r♦✈❡❞ ❜② ❈✇✐❦❡• ❛♥❞ ◆✐•ss♦♥ ✐♥ ❬✽(✳ ❈❤❛♣t❡r ✷ Pr❡•✐♠✐♥❛r✐❡s ✐♥ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ■♥t❡❣r❛t✐♦♥ ❚❤❡♦r② ❚❤✐s ❝❤❛♣t❡r ✐s ♠❡❛♥t ❛s ❛ t♦♦•❜♦① ✐♥ ✈✐❡✇ ♦❢ t❤❡ t✇♦ ❢♦••♦✇✐♥❣ ♦♥❡s✳ ■♥ t❤❡ ✜rst s❡❝t✐♦♥✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❜❛s✐❝ ❞❡✜♥✐t✐♦♥s ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ✐♥t❡❣r❛t✐♦♥✳ ■♥ t❤❡ s❡❝♦♥❞✱ ✇❡ r❡❝❛•• ❛♥❞ ♣r♦✈❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ K✲❢✉♥❝t✐♦♥❛• ❢♦r ❝♦✉♣•❡s ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ L p ✲s♣❛❝❡s✳ ■♥ t❤❡ t❤✐r❞✱ ✇❡ ✐♥tr♦❞✉❝❡ ✉•tr❛♣r♦❞✉❝ts ❛♥❞ ♣r♦✈❡ s♦♠❡ ♦❢ t❤❡✐r ♣r♦♣❡rt✐❡s✳ ❋✐♥❛••②✱ ✇❡ ❞❡t❛✐• t❤❡ ♣r♦♦❢ ♦❢ ❛♥ ✐♥❡q✉❛•✐t② ✇❤✐❝❤ ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❬✺✹( ❛♥❞ ✇✐•• ❜❡ ❝r✉❝✐❛• ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳ ✷✳✶ ❉❡✜♥✐t✐♦♥ ❛♥❞ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ♥♦♥❝♦♠✲ ♠✉t❛t✐✈❡ L p ✲s♣❛❝❡s ✷✳✶✳✶ ❉❡✜♥✐t✐♦♥ ❆ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ ✐s t❤❡ ❞❛t❛ ♦❢ ❛ ✈♦♥ ◆❡✉♠❛♥♥ ❛•❣❡❜r❛ M ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ s❡♠✐✜♥✐t❡ ♥♦r♠❛• ❢❛✐t❤❢✉• tr❛❝❡ τ ✳ ❆♥② ❡•❡♠❡♥t x ✐♥ M ❛❞♠✐ts ❛ ♣♦•❛r ❞❡❝♦♠♣♦s✐t✐♦♥ x = u |x| ✇❤❡r❡ u ✐s ❛ ✉♥✐t❛r② ❛♥❞ |x| ✐s ♣♦s✐t✐✈❡✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ |x| = (x * x) 1/2 ✳ ❋♦r ❛♥② p ∈ (0, ∞)✱ ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ L p ✲♥♦r♠ ✭♦r q✉❛s✐✲♥♦r♠✮ ♦♥ t❤✐s s♣❛❝❡ ❜② ❇♦r❡• ❢✉♥❝t✐♦♥❛• ❝❛•❝✉•✉s ❛♥❞ t❤❡ ❢♦••♦✇✐♥❣ ❢♦r♠✉•❛✿ x p = τ (|x| p ) 1/p . ❚❤❡ ❝♦♠♣•❡t✐♦♥ ♦❢ {x ∈ M : x p < ∞} ✇✐t❤ r❡s♣❡❝t t♦ . p ✐s ❞❡♥♦t❡❞ ❜② L p (M) ❛♥❞ ✈❡r✐✜❡s ♣r♦♣❡rt✐❡s s✐♠✐•❛r t♦ t❤♦s❡ ♦❢ ❝•❛ss✐❝❛• L p ✲s♣❛❝❡s✳ ❲❡ r❡❢❡r t♦ ❬✺✵( ❢♦r ❛ ♠♦r❡ ❝♦♠♣•❡t❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ L p ✲s♣❛❝❡s✳ ❚❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❛♥❛•♦❣ ♦❢ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥s ✐s ❞❡♥♦t❡❞ ❜② L 0 (M) ❛♥❞ ✐s t❤❡ s♣❛❝❡ ♦❢ ✉♥❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs x ❛✣•❛t❡❞ ✇✐t❤ M ✇❤✐❝❤ ❛r❡ τ ✲ ♠❡❛s✉r❛❜•❡ ✭✐✳❡ t❤❡r❡ ❡①✐sts λ ∈ R + s✉❝❤ t❤❛t τ (1 (λ,∞) (|x|)) < ∞✱ ✇❤❡r❡ ✹✺ 1 (λ,∞) (|x|) ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❇♦r❡• ❢✉♥❝t✐♦♥❛• ❝❛•❝✉•✉s✮✳ ❚❤✐s s♣❛❝❡ L 0 (M) ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♠❡❛s✉r❡ t♦♣♦•♦❣② ❝♦♥t✐♥✉♦✉s•② ❝♦♥t❛✐♥s L p (M) ❢♦r ❛•• p✳ ❚❤✐s ♠❛❦❡s ❛♥② ❝♦✉♣•❡ (L p (M)✱ L q (M)) ❝♦♠♣❛t✐❜•❡ ✐♥ t❤❡ s❡♥s❡ ♦❢ ✐♥t❡r♣♦•❛✲ t✐♦♥ ❬✻✵(✳ ❚❤❡ s✉♣♣♦rt ♦❢ ❛♥② s❡•❢✲❛❞•♦✐♥t ❡•❡♠❡♥t x ∈ L 0 (M) ✐s ❞❡✜♥❡❞ ❛s t❤❡ •❡❛st ♣r♦•❡❝t✐♦♥ s(x) s✉❝❤ t❤❛t s(x)x = x✳ ▲❡t M c = {x ∈ M : τ (s(|x|)) < ∞} ❜❡ t❤❡ s♣❛❝❡ ♦❢ ❜♦✉♥❞❡❞✱ ✜♥✐t❡•② s✉♣♣♦rt❡❞ ♦♣❡r❛t♦rs ✐♥ M✳ ❋♦r ❛♥② p ∈ (0, ∞)✱ M c ✐s ❞❡♥s❡ ✐♥ L p (M)✳ ❉❡♥♦t❡ ❜② P(M) t❤❡ s❡t ♦❢ ♦rt❤♦❣♦♥❛• ♣r♦•❡❝t✐♦♥s ✐♥ M ❛♥❞ ❜② P c (M) t❤❡ s❡t ♦❢ ✜♥✐t❡ ♣r♦•❡❝t✐♦♥s ✐♥ M✳ ✷✳✶✳✷ •❡♥❡r❛•✐s❡❞ s✐♥❣✉•❛r ✈❛•✉❡s ❆ ❝r✉❝✐❛• t♦♦• t♦ ✉♥❞❡rst❛♥❞ ❛♥❞ ♥♦♥❝♦♠♠✉t❛t✐✈❡ L p ✲s♣❛❝❡s ✐s t❤❡ ❣❡♥❡r❛•✐s❡❞ s✐♥❣✉•❛r ♥✉♠❜❡rs µ(x) ❛ss♦❝✐❛t❡❞ t♦ ❛♥② x ∈ L 0 (M)✳ ❚❤❡② ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② t❤❡ ❢♦••♦✇✐♥❣ ❢♦r♠✉•❛ ❬✶✻(✿ µ(x) :R + → R + t → µ t (x) = inf{ ex ∞ : e ∈ P(M), τ (1 -e) ≤ t}, = sup{a ∈ R + : τ (1 (a,∞) (|x|)) ≥ t}. ❚❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡s❡ ❢♦r♠✉•❛s ♠❛② ♥♦t ❜❡ ✐♠♠❡❞✐❛t❡•② ❝•❡❛r ❜✉t µ(x) ✐s t♦ ❜❡ t❤♦✉❣❤t ❛s ❛ ♥♦♥✲✐♥❝r❡❛s✐♥❣ ♥♦♥✲♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ❤❛s t❤❡ s❛♠❡ ❞✐str✐✲ ❜✉t✐♦♥ ❛s x✱ ✐♥ ♣❛rt✐❝✉•❛r µ(x) p = x p ❢♦r ❛•• p✳ ❚❤❡ ❣❡♥❡r❛•✐s❡❞ s✐♥❣✉•❛r ✈❛•✉❡s s❛t✐s❢② t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ Pr♦♣♦s✐t✐♦♥ ✷✳✶✳✶✳ ▲❡t x, y ∈ L 0 (M)✱ t, s > 0✱ t❤❡♥✿ ✶✳ ∀p ∈ (0, ∞)✱ ✐❢ x ✐s ♣♦s✐t✐✈❡✱ µ(x p ) = µ(x) p ✱ ✷✳ µ s+t (x + y) ≤ µ s (x) + µ t (y)✱ ✸✳ µ(x) ✐s r✐❣❤t✲❝♦♥t✐♥✉♦✉s✳ ❚❤❛♥❦s t♦ t❤❡ ❣❡♥❡r❛•✐s❡❞ s✐♥❣✉•❛r ✈❛•✉❡s✱ ✈❛r✐♦✉s ❝•❛ss✐❝❛• ♥♦t✐♦♥s ❝❛♥ ❜❡ ♥❛t✉r❛••② tr❛♥s•❛t❡❞ t♦ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❝♦♥t❡①t✳ ❚❤❡ ✐♥t❡r♣♦•❛t✐♦♥ ♦❢ ♥♦♥✲ ❝♦♠♠✉t❛t✐✈❡ L p ✲s♣❛❝❡s ❢♦r ❡①❛♠♣•❡✱ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ t❤❡ ❝•❛ss✐❝❛• ❝❛s❡ ✭s❡❡ r❡♠❛r❦ ✷✳✷✳✼✮✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ L p ✲s♣❛❝❡s ❝❛♥ ❛•s♦ ❜❡ ❣❡♥✲ ❡r❛•✐s❡❞ t♦ ❛♥② s②♠♠❡tr✐❝ s♣❛❝❡ ❬✸✶( ✳ ▲❡t E ❜❡ ❛ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡ ♦♥ (0, ∞)✳ ❉❡✜♥❡✿ E(M) = {x ∈ L 0 (M) : µ(x) ∈ E}. E(M) ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ✭q✉❛s✐✲✮♥♦r♠ x E(M) = µ(x) E ✐s ❛ ✭q✉❛s✐✲✮❇❛♥❛❝❤ s♣❛❝❡✳ ❲❡ s❛② t❤❛t E ✐s ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s②♠♠❡tr✐❝ s♣❛❝❡✳ ✷✳✶✳✸ ❚❡♥s♦r ♣r♦❞✉❝t ▲❡t (M 1 , τ 1 ) ❛♥❞ (M 2 , τ 2 ) ❜❡ t✇♦ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡s✱ ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡✐r t❡♥s♦r ♣r♦❞✉❝t (M, τ ) ✐♥ ❛ ♥❛t✉r❛• ✇❛②✳ ❋✐rst✱ ❝♦♥s✐❞❡r t❤❡ ❛•❣❡❜r❛✐❝ t❡♥s♦r ♣r♦❞✉❝t M 1 ⊗M 2 ✳ ■t ✐s ♥❛t✉r❛••② ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ tr❛❝❡ τ := τ 1 ⊗τ 2 ✇❤✐❝❤ ✐♥❞✉❝❡s ❛ s❝❛•❛r ♣r♦❞✉❝t ♦♥ M 1 ⊗ M 2 ✳ ❚❤✐s ❛••♦✇s✱ ❜② t❛❦✐♥❣ t❤❡ ❝♦♠♣•❡t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♥♦r♠ ❛ss♦❝✐❛t❡❞ t♦ t❤✐s t❡♥s♦r ♣r♦❞✉❝t✱ t♦ ❞❡✜♥❡ ❛ ❍✐•❜❡rt s♣❛❝❡ L 2 (M) ♦♥ ✇❤✐❝❤ M 1 ⊗ M 2 ❛❝ts ❜② ♠✉•t✐♣•✐❝❛t✐♦♥ ♦♥ t❤❡ r✐❣❤t✳ ❉❡✜♥❡ M t♦ ❜❡ t❤❡ ✇❡❛❦ ❝•♦s✉r❡ ♦❢ M 1 ⊗ M 2 ✐♥ B(L 2 (M))✱ τ 1 ⊗ τ 2 ❡①t❡♥❞s ✉♥✐q✉❡•② t♦ ❛ ♣♦s✐t✐✈❡ s❡♠✐✜♥✐t❡ ♥♦r♠❛• tr❛❝❡ τ ♦♥ M✳ ■♥ ❣❡♥❡r❛•✱ t❤❡ tr❛❝✐❛• t❡♥s♦r ♣r♦❞✉❝t ♦❢ M 1 ❛♥❞ M 2 ✐s ❞❡♥♦t❡❞ ❜② M 1 ⊗M 2 ✳ ❚❤✐s ❝♦♥str✉❝t✐♦♥ ♣r♦✈✐❞❡s ❛ ❣❡♥❡r❛• s❡tt✐♥❣ ❢♦r ✈❛r✐♦✉s ♦❜•❡❝ts t❤❛t ✇✐•• ❛♣♣❡❛r ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ❝❤❛♣t❡rs✳ ❋♦r ❡①❛♠♣•❡✱ s♦♠❡ s♣❛❝❡s ♦❢ ❢✉♥❝t✐♦♥s t❛❦✲ ✐♥❣ ✈❛•✉❡s ✐♥ ♥♦♥❝♦♠♠✉t❛t✐✈❡ L p ✲s♣❛❝❡s ❝❛♥ ❜❡ s❡❡♥ ❛s ♥♦♥❝♦♠♠✉t❛t✐✈❡ L p ✲ s♣❛❝❡s t❤❡♠s❡•✈❡s✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ❢♦r p ∈ (0, ∞)✱ ❛♥❞ n ∈ N✱ t❤❡ s♣❛❝❡s L p (R n , L p (M)) ✐s ❝❛♥♦♥✐❝❛••② ✐s♦♠❡tr✐❝ t♦ L p (M⊗L ∞ (R n ))✳ ✷✳✶✳✹ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ♠❛rt✐♥❣❛•❡s ▼❛rt✐♥❣❛•❡s ❛❞♠✐t ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❝♦✉♥t❡r♣❛rt✳ ❲❡ st❛rt ❜② ❞❡✜♥✐♥❣ ❝♦♥✲ ❞✐t✐♦♥❛• ❡①♣❡❝t❛t✐♦♥s✳ ▲❡t (M, τ ) ❜❡ ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ ❛♥❞ A ❛ s✉❜❛•❣❡❜r❛ ♦❢ M✳ ■❢ τ ✐s s❡♠✐✜♥✐t❡ ✇❤❡♥ r❡str✐❝t❡❞ t♦ A t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♥❞✐t✐♦♥❛• ❡①♣❡❝t❛t✐♦♥ E : M → A✳ E ✐s ✉♥✐t❛•✱ ♣♦s✐t✐✈❡✱ tr❛❝❡ ♣r❡s❡r✈✐♥❣ ❛♥❞ ❢♦r ❛♥② a, b ∈ A ❛♥❞ x ∈ M✱ E(axb) = aE(x)b✳ ■t ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s t❤❡ ❛❞•♦✐♥t ♦❢ t❤❡ ✐♥❝•✉s✐♦♥ ♠❛♣ i : L 1 (A) → L 1 (M)✳ ◆♦t❡ t❤❛t ❢♦r ❛•• p ≥ 1✱ E ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ❝♦♥tr❛❝t✐♦♥ ❢r♦♠ L p (M) t♦ L p (A)✳ ❚❤✐s ❢❛❝t ✐s ♥♦ •♦♥❣❡r tr✉❡ ✇❤❡♥ p < 1✳ ❆ ✜•tr❛t✐♦♥ F ♦♥ M ✐s ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ✭♣♦t❡♥t✐❛••② t✇♦✲s✐❞❡❞✮ ♦❢ s✉❜❛•❣❡❜r❛s (M n ) n∈N s✉❝❤ t❤❛t ❢♦r ❛•• n ∈ N✱ τ r❡str✐❝t❡❞ t♦ M n ✬s ✐s s❡♠✐✜♥✐t❡ ❛♥❞ n M n ✐s ✇❡❛❦•② ❞❡♥s❡ ✐♥ M✳ ❉❡♥♦t❡ ❜② E n t❤❡ ❝♦♥❞✐t✐♦♥❛• ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ M n ✳ ▲❡t p ∈ [1, ∞]✱ ❛ ♠❛rt✐♥❣❛•❡ ✐♥ L p (M)✱ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✜•tr❛t✐♦♥ F ✐s ❛ s❡q✉❡♥❝❡ (x n ) n∈N s✉❝❤ t❤❛t ❢♦r ❛•• n ∈ N✱ x n ∈ L p (M n ) ❛♥❞ E n (x n ) = x n-1 ✳ ❲❡ s❛② t❤❛t ❛ ♠❛rt✐♥❣❛•❡ ❝♦♥✈❡r❣❡s ✐♥ L p (M) ✐❢ t❤❡ s❡q✉❡♥❝❡ x n ❝♦♥✈❡r❣❡s t♦ ❛♥ ❡•❡♠❡♥t x ∞ ✐♥ L p (M)✳ ◆♦t❡ t❤❛t ✐♥ t❤✐s ❝❛s❡✱ E n (x ∞ ) = x n ✳ ❲❡ s❛② t❤❛t (x n ) n∈N ✐s ❜♦✉♥❞❡❞ ✐♥ L p ✐❢ sup n x n p < ∞✳ ❙✐♠✐•❛r•② t♦ t❤❡ ❝•❛ss✐❝❛• ❝❛s❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✿ Pr♦♣♦s✐t✐♦♥ ✷✳✶✳✷✳ ▲❡t p ∈ (1, ∞)✳ ❊✈❡r② ❜♦✉♥❞❡❞ ♠❛rt✐♥❣❛•❡ ✐♥ L p (M) ❝♦♥✲ ✈❡r❣❡s✳ ❋✐♥❛••②✱ ✐❢ x ✐s ❛ ♠❛rt✐♥❣❛•❡✱ ✇❡ ❞❡♥♦t❡ ❜② (dx n ) = (x n -x n-1 ) t❤❡ s❡q✉❡♥❝❡ ♦❢ ♠❛rt✐♥❣❛•❡ ❞✐✛❡r❡♥❝❡s ♦❢ x✳ ❲❡ ✇✐•• ❛•s♦ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ ∆ n = E n -E n-1 ✳ ✷✳✶✳✺ |♦✇s ❛♥❞ ❝♦•✉♠♥s ■♥ ❝•❛ss✐❝❛• ❤❛r♠♦♥✐❝ ❛♥❛•②s✐s✱ ❛ ❝♦♥s✐❞❡r❛❜•❡ ❛♠♦✉♥t ♦❢ ❛tt❡♥t✐♦♥ ❤❛s ❜❡❡♥ ❣✐✈❡♥ t♦ sq✉❛r❡ ❢✉♥❝t✐♦♥s✳ ❚❤❡② ❛♣♣❡❛r ❢♦r ❡①❛♠♣•❡ ✐♥ ▲✐tt•❡✇♦♦❞✲P❛•❡② t❤❡♦r② ❛♥❞ ✹✽❈❍❆P❚❊| ✷✳ P|❊▲■▼■◆❆|■❊❙ ■◆ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ■◆❚❊•|❆❚■❖◆ ❚❍❊❖|( ✐♥ t❤❡ t❤❡♦r② ♦❢ ❢✉♥❝t✐♦♥❛• ❝❛•❝✉•✉s ❢♦r s❡♠✐❣r♦✉♣s✳ ❚❤❡s❡ ❛r❡ ❡①♣r❡ss✐♦♥s ♦❢ t❤❡ ❢♦r♠ i∈N |f i | 2 1/2 ✳ ■♥ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s❡tt✐♥❣ ❤♦✇❡✈❡r✱ ✐t ✐s ♥♦t ❝•❡❛r ❛t ✜rst ❤♦✇ t♦ ❛ss♦❝✐❛t❡ ❛ sq✉❛r❡ ❢✉♥❝t✐♦♥ t♦ ❛ s❡q✉❡♥❝❡ (x i ) i∈N ∈ M N ✳ ■♥❞❡❡❞✱ t❤❡ ♥❛✐✈❡ ❡①♣r❡ss✐♦♥ i∈N |x i | 2 1/2 ❤❛s ♥♦ r❡❛s♦♥ t♦ ❜❡ ❝❤♦s❡♥ ♦✈❡r i∈N |x * i | 2 1/2 ❜✉t t❤❡ t✇♦ ❞♦ ♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥ ✐♥ ❣❡♥❡r❛•✳ ❚❤❡② ❝❛♥✱ ✐♥ ❢❛❝t✱ ✐❢ M ✐s ❛ ❢❛❝t♦r✱ ❜❡ ❛♥② t✇♦ ♣♦s✐t✐✈❡ ❡•❡♠❡♥ts ✇✐t❤ t❤❡ s❛♠❡ ♥♦r♠ ✐♥ L 2 ✭❛s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❬✺✾(✱ t❤❡♦r❡♠ ✸✳✺✮✳ ❚❤❡ ❛♥s✇❡r t♦ ❝♦♥str✉❝t✐♥❣ ♥♦♥❝♦♠♠✉t❛t✐✈❡ sq✉❛r❡ ❢✉♥❝t✐♦♥s ✐s ❣✐✈❡♥ ❜② t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✇❤✐❝❤ ✇✐•• ❜❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✳ ❋♦r ♥♦✇✱ ✇❡ ✇✐•• s✐♠♣•② ✐♥tr♦❞✉❝❡ s♦♠❡ ✉s❡❢✉• t❡r♠✐♥♦•♦❣②✳ ❉❡♥♦t❡ ❜② S(M) t❤❡ s❡t ♦❢ s❡q✉❡♥❝❡s ♦❢ ❡•❡♠❡♥ts ♦❢ M c ✳ ❋♦r a, b ∈ M ❛♥❞ x ∈ S(M) ✇❡ ❝❛♥ ♥❛t✉r❛••② ❞❡✜♥❡ t❤❡ ♣r♦❞✉❝t axb ❜②✿ axb = (ax n b) n≥0 . ▲❡t x ∈ S(M)✳ ❉❡♥♦t❡ ❜② N ∈ N ✐ts •❡♥❣t❤ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ♣r♦•❡❝t✐♦♥✱ e = s N i=0 x * i x i ∨ s N i=0 x i x * i . ❚❤❡ ❢❛❝t t❤❛t x ∈ (eMe) N ✇✐•• ❜❡ ✉s❡❢✉• ✐♥ ✈❛r✐♦✉s ♣r♦♦❢s ✐♥ t❤❡ ♥❡①t ❝❤❛♣✲ t❡r s✐♥❝❡ ✐t ❡♥❛❜•❡s ✉s t♦ ✇♦r❦ ✇✐t❤ ♦♥•② ✜♥✐t❡ s❡q✉❡♥❝❡s ♦❢ ✜♥✐t❡•② s✉♣♣♦rt❡❞ ❡•❡♠❡♥ts✳ |❡❝❛•• t❤❛t t❤❡ ❡•❡♠❡♥ts ♦❢ B(ℓ 2 ) ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ ✐♥✜♥✐t❡ ♠❛tr✐❝❡s ❛♥❞ t❤❛t B(ℓ 2 ) ✐s ❡♥❞♦✇❡❞ ✇✐t❤ ✐ts ❝❛♥♦♥✐❝❛• tr❛❝❡✳ ❲❡ ✇✐•• ❞❡♥♦t❡ ❜② e n,m t❤❡ ❡•❡♠❡♥t (δ i,n δ j,m ) i,j∈N ∈ B(ℓ 2 )✳ ❋♦r x ✐♥ S(M)✱ ✇❡ ❞❡✜♥❡ Rx = n≥0 x n ⊗e 1,n ✱ Cx = n≥0 x n ⊗ e n,1 ✇❤✐❝❤ ❛r❡ ✉♥❞❡rst♦♦❞ ❛s ❡•❡♠❡♥ts ♦❢ t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛•❣❡❜r❛ M B := M⊗B(ℓ 2 )✳ ❇♦t❤ ❛•❣❡❜r❛s ❛r❡ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ t❡♥s♦r ♣r♦❞✉❝t tr❛❝❡✳ ❲❤❡♥ ✐t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ✉s✱ M ✇✐•• ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ M ⊗ e 1,1 ✐♥ M B ✳ ❲✐t❤ t❤✐s ✐♥ ♠✐♥❞✱ ♥♦t❡ t❤❛t t❤❡ t✇♦ ✧sq✉❛r❡ ❢✉♥❝t✐♦♥s✧ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❛s t❤❡ ♠♦❞✉•✐ ♦❢ Rx ❛♥❞ Cx✿ |(Rx) * | = n≥0 x n x * n 1/2 ❛♥❞ |Cx| = n≥0 x * n x n 1/2 ✳ ❋✐① E ❛ s②♠♠❡tr✐❝ s♣❛❝❡✳ ❉❡✜♥❡ x R E := Rx E(M B ) ✭r❡s♣✳ x C E := Cx E(M B ) ✮ ❛♥❞ R E ✭r❡s♣✳ C E ✮ t❤❡ ❝♦♠♣•❡t✐♦♥ ♦❢ S(M) ✇✐t❤ r❡s♣❡❝t t♦ . R E ✭r❡s♣✳ . C E ✮✳ ❚♦ •✐❣❤t❡♥ t❤❡ ♥♦t❛t✐♦♥s✱ ❢♦r p ∈ (0, ∞]✱ ✇❡ ✇r✐t❡ C p = C Lp ❛♥❞ R p = R Lp ✳ ◆♦t❡ t❤❛t R p ❛♥❞ C p ❛r❡ ❝♦♥t✐♥✉♦✉s•② ✐♥❝•✉❞❡❞ ✐♥ L 0 (M) N ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s♣❛❝❡s R p + C p ❛♥❞ R p ∩ C p ❛r❡ ✇❡••✲❞❡✜♥❡❞✳ ❚❤❡② ❜♦t❤ ❝♦♥t❛✐♥ S(M) ❛s ❛ ❞❡♥s❡ s✉❜s♣❛❝❡ ✭✇❡❛❦✲ * ❞❡♥s❡ ❢♦r p = ∞✮✳ |❡♠❛r❦ ✷✳✶✳✸✳ ◆♦t❡ t❤❛t R r ❛♥❞ C r ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥t♦ L r (M B ) ❜② ❡①t❡♥❞✲ ✐♥❣ R ❛♥❞ C✳ ❚❤❡② ❝❛♥ ❜② t❤✐s ♠❡❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ ❝•♦s❡❞✱ 1✲❝♦♠♣•❡♠❡♥t❡❞ s✉❜s♣❛❝❡s ♦❢ L r (M B )✳ ❚❤✐s ❛•s♦ ♠❛❦❡s t❤❡ ❝♦✉♣•❡s (R p , R q ) ❛♥❞ (C p , C q ) ❝♦♠✲ ♣❛t✐❜•❡ ✐♥ t❤❡ s❡♥s❡ ♦❢ ✐♥t❡r♣♦•❛t✐♦♥✳ ✷✳✷✳ ❙❖▼❊ P|❖P❊|❚■❊❙ ❖❋ ❚❍❊ K✲❋❯◆❈❚■❖◆❆▲ ✹✾ ▲❡t ✉s ♥♦✇ ♣r♦✈❡ ❛ •❡♠♠❛ t❤❛t ✇✐•• ❜❡ ✉s❡❢✉• ✇❤❡♥ ♠❛♥✐♣✉•❛t✐♥❣ r♦✇s ❛♥❞ ❝♦•✉♠♥s✳ ▲❡♠♠❛ ✷✳✶✳✹✳ ▲❡t p > 0✱ y, z ∈ S(M) ❛♥❞ e, f ∈ P(M) s✉❝❤ t❤❛t e ❝♦♠♠✉t❡s ✇✐t❤ |(Ry) * |✳ ❚❤❡♥✱ ✐✳ e |(Ry) * | p = |(R(ey)) * | p , ✐✐✳ Ry p p = R(ey) p p + R(e ⊥ y) p p , ✐✐✐✳ |C(f z)| 2 ≤ |Cz| 2 ❛♥❞ ❝♦♥s❡q✉❡♥t•② C(f z) p ≤ Cz p . Pr♦♦❢✳ ✐✳ ❚❤✐s ✐s ❛ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✿ e |(Ry) * | p = e p/2 (|(Ry) * | 2 ) p/2 = (e |(Ry) * | 2 ) p/2 = (e( i y i y * i )e) p/2 = ( i (ey i )(ey i ) * ) p/2 = |R(ey) * | p . ✐✐✳ ❲r✐t❡ f := e ⊥ ✳ ❯s✐♥❣ ✐✳✱ t❤✐s ✐s ❛❣❛✐♥ ❛ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✿ Ry p p = τ (|(Ry) * | p ) = τ ((e + f ) |(Ry) * | p ) = τ (e |(Ry) * | p ) + τ (f |(Ry) * | p ) = τ (|(R(ey)) * | p ) + τ (|(R(f y)) * | p ) = R(ey) p p + R(f y) p p . ✐✐✐✳ ■♥❞❡❡❞✱ |C(f z)| 2 = i z * i f z i ≤ i z * i z i = |Cz| 2 . ✷✳✷ ❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ K✲❢✉♥❝t✐♦♥❛• ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ✐♥t❡❣r❛t✐♦♥✱ t❤❡ K✲❢✉♥❝t✐♦♥❛• ❞♦❡s ♥♦t ❞❡✲ ♣❡♥❞✱ ✉♣ t♦ ✉♥✐✈❡rs❛• ❝♦♥st❛♥ts✱ ♦♥ t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛•❣❡❜r❛ ✐♥ ✇❤✐❝❤ ✐t ✐s ❝❛•❝✉•❛t❡❞✳ ❚❤✐s r❡s✉•t ✇❛s ♣r♦✈❡❞ ❜② ❳✉ ✐♥ ❤✐s ✉♥♣✉❜•✐s❤❡❞ •❡❝t✉r❡ ♥♦t❡s ✭❬✻✷(✮✳ ❲❡ ✇✐•• ❣✐✈❡ ❛♥ ❛•t❡r♥❛t✐✈❡ ♣r♦♦❢ ❤❡r❡ ✉s✐♥❣ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ✇❤✐❝❤ ✇❡ ✇✐•• ❛•s♦ ♥❡❡❞ ✐♥ t❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤❡ t❡①t✳ ■t ✐s ❛ ✈❡rs✐♦♥ ♦❢ t❤❡ ♣♦✇❡r t❤❡♦r❡♠ ✭s❡❡ ❬✶(✮ ✐♥ t❤❡ ♣❛rt✐❝✉•❛r ❝❛s❡ ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ L p ✲s♣❛❝❡s✳ Pr♦♣♦s✐t✐♦♥ ✷✳✷✳✶✳ ▲❡t α > 0✱ x ∈ (L p (M) + L q (M)) + ❛♥❞ p, q ∈ (0, ∞] t❤❡♥✿ K t (x, L p (M), L q (M)) ≤ c p,α K t 1/α (x 1/α , L pα (M), L qα (M)) α , ✇❤❡r❡ c p,α = max(1, 2 1/pα-1 ) max(2 α-1 , 2 1-α )✳ |❡♠❛r❦ ✷✳✷✳✷✳ ❲❡ ✇✐•• ✉s❡ t❤✐s ♣r♦♣♦s✐t✐♦♥ ✉♥❞❡r t❤❡ ❢♦••♦✇✐♥❣ ❢♦r♠✳ ▲❡t α > 0✱ x ∈ (L p (M) + L q (M)) + ❛♥❞ p, q ∈ (0, ∞] t❤❡♥✿ K t (x, L p (M), L q (M)) ≈ K t 1/α (x 1/α , L pα (M), L qα (M)) α , ✺✵❈❍❆P❚❊| ✷✳ P|❊▲■▼■◆❆|■❊❙ ■◆ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ■◆❚❊•|❆❚■❖◆ ❚❍❊❖|( ❲❡ ✉s❡ t❤❡ ❢♦••♦✇✐♥❣ r♦✉t✐♥❡ ♦♣❡r❛t♦r ✐♥❡q✉❛•✐t✐❡s✱ ♣r♦✈❡❞ ❛t t❤❡ ❡♥❞ ♦❢ t❤✐s s❡❝t✐♦♥✳ ▲❡♠♠❛ ✷✳✷✳✸✳ ▲❡t a, b ∈ L 0 (M) + ✱ α ≥ 1 ❛♥❞ θ ≤ 1✳ ✐✳ ✐❢ 0 ≤ a ≤ b t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♥tr❛❝t✐♦♥ c s✉❝❤ t❤❛t a = cbc * ✱ ✐✐✳ ✐❢ 0 ≤ a ≤ b t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♣❛rt✐❛• ✐s♦♠❡tr② u s✉❝❤ t❤❛t a 2 ≤ ub 2 u * ✱ ✐✐✐✳ t❤❡r❡ ❡①✐sts ❛ ♣❛rt✐❛• ✐s♦♠❡tr② u ∈ M s✉❝❤ t❤❛t✿ (a + b) α ≤ 2 α-1 u(a α + b α )u * , ✐✈✳ t❤❡r❡ ❡①✐st t✇♦ ♣❛rt✐❛• ✐s♦♠❡tr✐❡s u ❛♥❞ v s✉❝❤ t❤❛t✿ (a + b) θ ≤ ua θ u * + vb θ v * . Pr♦♦❢ ♦❢ ♣r♦♣♦s✐t✐♦♥ ✷✳✷✳✶✳ ❋♦r ❛•• r > 0✱ ❞❡✜♥❡ a r := max(1, 2 1/r-1 )✳ ❚❛❦❡ a ∈ L pα (M) ❛♥❞ b ∈ L qα (M) s✉❝❤ t❤❛t a + b = x 1/α ✳ ❲❡ st❛rt ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❝❛s❡ α < 1✳ ❲❡ ♥❡❡❞ t♦ ✜♥❞ a ′ ∈ L p ❛♥❞ b ′ ∈ L q s✉❝❤ t❤❛t a ′ + b ′ = x ❛♥❞✿ a ′ p + t b ′ q ≤ a pα 2 1-α a pα + t 1/α b qα α . ❙✐♥❝❡ x ✐s ♣♦s✐t✐✈❡✱ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t a ❛♥❞ b ❛r❡ ♣♦s✐t✐✈❡✳ ■♥❞❡❡❞✱ ✜rst ♥♦t❡ t❤❛t✿ x = a + a * 2 + b + b * 2 , ❛♥❞✿ a + a * 2 pα + t b + b * 2 qα ≤ a pα ( a pα + t b qα ) ✉s✐♥❣ t❤❡ tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t② ❢♦r pα ≥ 1 ❛♥❞ t❤❡ pα✲tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t② ❢♦r pα ≤ 1✳ ❙♦ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t a ❛♥❞ b ❛r❡ s❡•❢❛❞•♦✐♥t ❛♥❞ ✇r✐t❡ a = a + -a - ❛♥❞ b = b + -b -t❤❡✐r ❞❡❝♦♠♣♦s✐t✐♦♥s ✐♥t♦ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ♣❛rts✳ ■t ❢♦••♦✇s t❤❛t x ≤ a + + b + s♦ t❤❡r❡ ❡①✐sts ❛ ❝♦♥tr❛❝t✐♦♥ c s✉❝❤ t❤❛t x = ca + c * + cb + c * ✇❤✐❝❤ ②✐❡•❞s ❛ ❜❡tt❡r ❞❡❝♦♠♣♦s✐t✐♦♥ t❤❛♥ a, b✳ ❯s✐♥❣ •❡♠♠❛ 2.2.3✱ ✇❡ ❝❛♥ ✜♥❞ t✇♦ ❝♦♥tr❛❝t✐♦♥s u ❛♥❞ v s✉❝❤ t❤❛t✿ x = ua α u * + vb α v * =: a ′ + b ′ . ❆♥❞ ✇❡ ❤❛✈❡✿ a ′ p + t b ′ q ≤ a α p + b α q = a α pα + t α b α qα ≤ 2 1-α a pα + t b qα α . ❚❛❦✐♥❣ t❤❡ ✐♥✜♠✉♠ ♦✈❡r ❛•• ❞❡❝♦♠♣♦s✐t✐♦♥s x 1/α = a + b✱ ❛♥❞ r❡❝❛••✐♥❣ t❤❛t ✇❡ ❤❛❞ t♦ ❛❞❞ ❛ ❢❛❝t♦r a pα t♦ ❝♦♥s✐❞❡r ♦♥•② a ❛♥❞ b ♣♦s✐t✐✈❡✱ ✇❡ ♦❜t❛✐♥✿ K t (x, L p (M), L q (M)) ≤ a pα 2 1-α K t 1/α (x 1/α , L pα (M), L qα (M)) α . ✷✳✷✳ ❙❖▼❊ P|❖P❊|❚■❊❙ ❖❋ ❚❍❊ K✲❋❯◆❈❚■❖◆❆▲ ✺✶ ▲❡t ✉s ♥♦✇ ❝♦♥s✐❞❡r α ≥ 1✳ ❈♦♥t✐♥✉❡ t♦ ❛ss✉♠❡ a ❛♥❞ b t♦ ❜❡ ♥♦♥♥❡❣❛t✐✈❡ ❛♥❞ r❡❝❛•• t❤❛t ✇❡ •♦s❡ ❛ ❝♦♥st❛♥t a pα ❜② ❞♦✐♥❣ s♦✳ ❯s✐♥❣ •❡♠♠❛ 2.2.3 t❤❡r❡ ❡①✐sts ❛ ❝♦♥tr❛❝t✐♦♥ u s✉❝❤ t❤❛t✿ x = 2 α-1 u(a α + b α )u * = 2 α-1 ua α u * + 2 α-1 ub α u * =: a ′ + b ′ . ◆♦✇✱ ✇❡ ❝♦♠♣✉t❡✿ a ′ p + t b ′ q ≤ 2 α-1 ( a α pα + t b α qα ) ≤ 2 α-1 ( a pα + t 1/α b qα ) α . ❚❛❦✐♥❣ t❤❡ ✐♥✜♠✉♠ ♦✈❡r ❛•• ❞❡❝♦♠♣♦s✐t✐♦♥s✱ t❤❡ ♣r♦♦❢ ✐s ❝♦♠♣•❡t❡✳ ❲❡ ✇✐•• ♥♦✇ ❜❡❣✐♥ t♦ ♣r♦✈❡ t❤❡ r❡s✉•t ♠❡♥t✐♦♥❡❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤✐s s❡❝t✐♦♥ ✐✳❡✳ t❤❛t t❤❡ K✲❢✉♥❝t✐♦♥❛• ♦❢ ❛♥ ❡•❡♠❡♥t ♦♥•② ❞❡♣❡♥❞s ♦♥ ✐ts ❞✐str✐❜✉t✐♦♥✳ ❚❤❡ ✜♥✐t❡ ❝❛s ❝❛♥ ❜❡ ♣r♦✈❡❞ ✇✐t❤♦✉t ♠✉❝❤ tr♦✉❜•❡ ❜② ✉s✐♥❣ ❝♦♥❞✐t✐♦♥❛• ❡①♣❡❝✲ t❛t✐♦♥s✳ ❋♦r t❤❡ s❡♠✐✜♥✐t❡ ❝❛s❡ ❤♦✇❡✈❡r✱ ✇❡ ♥❡❡❞ t❤❡ ❢♦••♦✇✐♥❣ t✇♦ •❡♠♠❛s✳ ▲❡♠♠❛ ✷✳✷✳✹✳ ▲❡t p, q ∈ (1, ∞] ❛♥❞ t > 0✳ ▲❡t x ∈ L p (M) + L q (M)✱ t❤❡♥ K t (x, L p (M), L q (M)) = sup e∈Pc(M) K t (exe, L p (M), L q (M)). Pr♦♦❢✳ ❉❡✜♥❡ E = L p ′ (M) ∩ t -1 L q ′ (M) ✇✐t❤ p ′ = (1 -p -1 ) -1 ❛♥❞ q ′ = (1 - q -1 ) -1 ❛♥❞ ♥♦t❡ t❤❛t L p (M) + tL q (M) = E * ✳ ❙✐♥❝❡ p ′ ❛♥❞ q ′ ❜❡•♦♥❣ t♦ [1, ∞)✱ M c ✐s ❞❡♥s❡ ✐♥ E✳ ❍❡♥❝❡✱ K t (x, L p (M), L q (M)) = sup{|τ (xy)| : y ∈ M c , y E = 1} = sup{|τ (xeye)| : e ∈ P c (M), y ∈ M c , y E = 1} = sup e∈Pc(M) exe Lp(M)+tLq(M) = sup e∈Pc(M) K t (exe, L p (M), L q (M)). ▲❡♠♠❛ ✷✳✷✳✺✳ ▲❡t x ∈ L + 0 (M)✱ ǫ > 0 ❛♥❞ f ∈ P c (L ∞ (0, ∞))✳ ❚❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ ♣r♦•❡❝t✐♦♥ e ∈ M s✉❝❤ t❤❛t✿ µ(µ(x)f ) ≤ µ(exe) + ε. Pr♦♦❢✳ ▲❡t f ∈ P c (L ∞ (0, ∞)) ❛♥❞ ❞❡✜♥❡ T := R + f ✱ t❤❡♥ µ(µ(x)f ) µ(x)1 (0,T ) = µ(µ(x)1 (0,T ) ). ❍❡♥❝❡✱ ✐t s✉✣❝❡s t♦ ❝♦♥s✐❞❡r ♣r♦•❡❝t✐♦♥s ♦❢ t❤❡ ❢♦r♠ 1 (0,T ) ✱ T > 0✳ ▲❡t a = µ T (x) ❛♥❞ •❡t e 1 = 1 (a,∞) (x)✳ ❇② ❞❡✜♥✐t✐♦♥ ♦❢ µ✱ e 1 ✐s ✜♥✐t❡✱ ✇r✐t❡ t 1 = τ (e 1 )✳ ◆♦t❡ t❤❛t µ t (x) = a ❢♦r t 1 ≤ t < T ✳ ❙✐♥❝❡ a = µ(T )✱ τ (1 (a-ε,∞) ) ≥ T ✱ ❤❡♥❝❡ τ (1 (a-ε,a] ) ≥ T -t 1 ✳ ▲❡t e 2 ❜❡ ❛ ✜♥✐t❡ ♣r♦•❡❝t✐♦♥ s✉❝❤ t❤❛t e 2 ≤ 1 (a-ε,a] ❛♥❞ τ (e 2 ) ≥ T -t 1 ✳ ■t ✐s ♣♦ss✐❜•❡ t♦ ✜♥❞ s✉❝❤ ❛ ♣r♦•❡❝t✐♦♥ s✐♥❝❡ τ ✐s s❡♠✐✜♥✐t❡✳ ❉❡✜♥❡ e = e 1 + e 2 ✳ ❇② ❝♦♥str✉❝t✐♦♥✱ ❢♦r t ∈ (0, t 1 )✱ µ t (exe) = µ t (x) = f (t)µ t (x) ❛♥❞ ❢♦r t ∈ [t 1 , T )✱ µ t (exe) ≥ a -ε = f (t)µ t (x) -ε✳ ❙♦ e s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ •❡♠♠❛✳ Pr♦♣♦s✐t✐♦♥ ✷✳✷✳✻✳ ▲❡t x ∈ L p (M) + L q (M)✱ p, q ∈ R s✉❝❤ t❤❛t 0 < p < q ❛♥❞ t > 0✿ K t (µ(x), L p (0, ∞), L q (0, ∞)) ≈ cp K t (x, L p (M), L q (M)), ✇❤❡r❡ c p ♦♥•② ❞❡♣❡♥❞s ♦♥ p✳ Pr♦♦❢ ♦❢ ♣r♦♣♦s✐t✐♦♥ ✷✳✷✳✻✳ ❙✉♣♣♦s❡ t❤❛t x ✐s ♣♦s✐t✐✈❡✳ ❲❡ ❞♦ ♥♦t •♦s❡ ❣❡♥❡r❛•✐t② ❤❡r❡ s✐♥❝❡ ♠✉•t✐♣•②✐♥❣ ❜② ❛ ♣❛rt✐❛• ✐s♦♠❡tr② ❞♦❡s ♥♦t ❝❤❛♥❣❡ t❤❡ K✲❢✉♥❝t✐♦♥❛•✳ ❋✐rst✱ ❛ss✉♠❡ t❤❛t p > 1✳ ▲❡t e ∈ P c (M) ❜❡ ❛ ✜♥✐t❡ ♣r♦•❡❝t✐♦♥✳ ▲❡t ✉s ♣r♦✈❡ t❤❡ ♣r♦♣♦s✐t✐♦♥ ❢♦r y = exe✳ ❚♦ ❞♦ s♦✱ ✇❡ ❝❛♥ ✇♦r❦ ✐♥ t❤❡ ✜♥✐t❡ ❛•❣❡❜r❛ eMe ✇❤✐❝❤ ❝•❡❛r•② ❞♦❡s ♥♦t ♠♦❞✐❢② t❤❡ K✲❢✉♥❝t✐♦♥❛• ♦❢ y✳ ❉❡♥♦t❡ ❜② M y t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛•❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② y ∈ M ✇❤✐❝❤ ✐s ❛❜❡•✐❛♥✳ ❚❤❡r❡ ❛r❡ t✇♦ ❝♦♥❞✐t✐♦♥❛• ❡①♣❡❝t❛t✐♦♥s E 1 : M → M y ❛♥❞ E 2 : L ∞ (0, τ (e)) → M µ(y) ❛♥❞ M y ✐s ❝❛♥♦♥✐❝❛••② ✐s♦♠♦r♣❤✐❝ t♦ M µ(y) ❜② y → µ(y)✳ ❙✐♥❝❡ ❝♦♥❞✐t✐♦♥❛• ❡①♣❡❝t❛t✐♦♥s ❡①t❡♥❞ t♦ ❝♦♥tr❛❝t✐♦♥s ♦♥ L p ✱ ✇❡ ❤❛✈❡✱ K t (y, L p (M), L q (M)) = K t (µ(y), L p (0, ∞), L q (0, ∞)). ▲❡t ✉s ❝♦♥s✐❞❡r x ♦♥❝❡ ❛❣❛✐♥✳ ❈♦♠❜✐♥✐♥❣ t❤❡ ❡q✉❛•✐t② ❛❜♦✈❡ ✇✐t❤ •❡♠♠❛ ✷✳✷✳✹✱ ✇❡ ♦❜t❛✐♥✿ K t (x, L p (M), L q (M)) = sup e∈Pc(M) K t (µ(exe), L p (0, ∞), L q (0, ∞)) ✭✷✳✶✮ ❛♥❞ K t (µ(x), L p (0, ∞), L q (0, ∞)) = sup f ∈Pc(L∞(0,∞)) K t (µ(f µ(x)f ), L p (0, ∞), L q (0, ∞)). ✭✷✳✷✮ ❙✐♥❝❡ ❢♦r ❛•• e ∈ P(M)✱ µ(exe) ≤ µ(x) ✭s❡❡ ❬✶✻(✮✿ K t (x, L p (M), L q (M)) ≤ K t (µ(x), L p (0, ∞, L q (0, ∞)). ❈♦♥✈❡rs❡•②✱ ❜② •❡♠♠❛ ✷✳✷✳✺✱ ❢♦r ❡✈❡r② ♣r♦•❡❝t✐♦♥ f ∈ P c (L ∞ (0, ∞)) ❛♥❞ ε > 0✱ t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ ♣r♦•❡❝t✐♦♥ e ✐♥ M✱ s✉❝❤ t❤❛t µ(exe) + ε ≥ µ(f µ(x))✳ ❚❤✐s ✐♠♣•✐❡s t❤❛t sup e∈Pc(M) K t (µ(exe), L p (0, ∞), L q (0, ∞)) ≥ sup f ∈Pc(L∞(0,∞)) K t (f µ(x)f, L p (0, ∞), L q (0, ∞)), ❛♥❞ ❡♥❛❜•❡s ✉s t♦ ❝♦♥❝•✉❞❡ ✉s✐♥❣ ✭✷✳✶✮ ❛♥❞ ✭✷✳✷✮✳ ◆♦✇ t❛❦❡ 0 < p ≤ 1 ❛♥❞ q > p✳ ❚❤❡ r❡s✉•t ❢♦••♦✇s ❢r♦♠ ♣r♦♣♦s✐t✐♦♥ 2.2.1✿ K t (y, L p (M), L q (M)) ≈ (K t p/2 (y p/2 , L 2 (M), L 2q/p (M)) 2/p = K t p/2 (µ(y p/2 ), L 2 (R + ), L 2q/p (R + )) 2/p ≈ K t (µ(y), L p (R + ), L q (R + )). ✷✳✷✳ ❙❖▼❊ P|❖P❊|❚■❊❙ ❖❋ ❚❍❊ K✲❋❯◆❈❚■❖◆❆▲ ✺✸ |❡♠❛r❦ ✷✳✷✳✼✳ ■♥ ❬✻✷(✱ t❤❡ r❡s✉•t ❛❜♦✈❡ ✐s ♦❜t❛✐♥❡❞ ❜② ✜rst ♣r♦✈✐♥❣ t❤❛t t❤❡ ❝♦✉♣•❡ (L 1 (M), L ∞ (M)) ✐s✱ ✐♥ ✐♥t❡r♣♦•❛t✐♦♥ •❛♥❣✉❛❣❡✱ ❛ ♣❛rt✐❛• r❡tr❛❝t ♦❢ t❤❡ ❝♦✉♣•❡ (L 1 (0, ∞), L ∞ (0, ∞))✳ ❚❤✐s ❡♥❛❜•❡s ✉s t♦ ❞❡✜♥❡✿ K t (x, p, q) := K t (µ(x), L p (R + ), L q (R + ))✳ |❡♠❛r❦ ✷✳✷✳✽✳ ❚❤❡ ❡q✉✐✈❛•❡♥❝❡ ❛❜♦✈❡ ❛•s♦ ❛♣♣•✐❡s t♦ t❤❡ r♦✇ ❛♥❞ ❝♦•✉♠♥ s♣❛❝❡s✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ❢♦r x ∈ S(M) ❛♥❞ p, q > 0✱ K t (x, R p , R q ) ≈ K t (Rx, p, q) ❛♥❞ K t (x, C p , C q ) ≈ K t (Cx, p, q)✳ Pr♦♦❢✳ ❯s✐♥❣ t❤❡ ❢❛❝t t❤❛t ❢♦r ❛•• r > 0✱ R r ❛♥❞ C r ❛r❡ ❝♦♠♣•❡♠❡♥t❡❞ ✐♥ L r (M⊗B(ℓ 2 )) ❛♥❞ ♣r♦♣♦s✐t✐♦♥ ✷✳✷✳✻✱ K t (x, R p , R q ) ≈ K t (Rx, L p (M⊗B(ℓ 2 )), L q (M⊗B(ℓ 2 ))) ≈ K t (Rx, p, q) ❛♥❞ s✐♠✐•❛r•② ❢♦r ❝♦•✉♠♥s✳ ❆s s❤♦✇♥ ✐♥ t❤❡ ♥❡①t ♣r♦♣♦s✐t✐♦♥✱ t❤❡ K✲❢✉♥❝t✐♦♥❛•✱ K t (x, p, ∞)✱ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❛ s✉♣r❡♠✉♠ ♦✈❡r ♣r♦•❡❝t✐♦♥s✳ ❚❤✐s ✐s ❝r✉❝✐❛• ❢♦r t❤❡ ♥❡①t ❝❤❛♣t❡r✳ Pr♦♣♦s✐t✐♦♥ ✷✳✷✳✾✳ ❙✉♣♣♦s❡ t❤❛t M ✐s ❞✐✛✉s❡✳ ▲❡t p > 0✳ ❚❤❡♥ ❢♦r ❛•• x ∈ M c ✿ sup{ ex p : τ (e) ≤ t p , e ∈ P(M)} ≈ Ap K t (x, p, ∞). Pr♦♦❢✳ ■❢ t p > τ (1)✱ ✇❡ ❤❛✈❡✿ sup{ ex p : τ (e) ≤ t p , e ∈ P(M)} = x p ❛♥❞ t❤❡ ♣r♦♣♦s✐t✐♦♥ ✐s ✈❡r✐✜❡❞ ❜② ♣r♦♣♦s✐t✐♦♥ ✶✳✸✳✷✳ ❙♦ ❢r♦♠ ♥♦✇ ♦♥✱ ❛ss✉♠❡ t❤❛t t p ≤ τ (1)✳ ❙✐♥❝❡ M ✐s ❞✐✛✉s❡✱ t❤❡r❡ ✐s ❛ ♣r♦•❡❝t✐♦♥ e ✐♥ M ✇✐t❤ tr❛❝❡ t p ✱ ❝♦♠♠✉t✐♥❣ ✇✐t❤ |x * |✱ s✉❝❤ t❤❛t✿ τ (e |x * | p ) = t p 0 µ s (x) p ds ≥ 1 A p K t (x, p, ∞) p . ❋✉rt❤❡r♠♦r❡✱ s✐♥❝❡ e ❛♥❞ |x * | •✐✈❡ ✐♥ ❛ ❝♦♠♠✉t❛t✐✈❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛•❣❡❜r❛✱ t❤❡② ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❢✉♥❝t✐♦♥s ✐♥ ❛ s♣❛❝❡ L ∞ (Ω)✱ t❤✉s✿ τ (e |x * | p ) = τ (e p/2 (xx * ) p/2 e p/2 ) = τ ((exx * e) p/2 ) = ex p p . ❍❡♥❝❡✿ K t (x, p, ∞) ≤ A p ex p ≤ A p sup{ ex p : τ (e) = t p , e ∈ P(M)}. ❚♦ ♣r♦✈❡ t❤❡ ❝♦♥✈❡rs❡ ✐♥❡q✉❛•✐t②✱ t❛❦❡ e ∈ P(M) s✉❝❤ t❤❛t τ (e) = t p ✳ ◆♦t❡ t❤❛t✿ ex p p = ∞ 0 µ(|ex| p ) = t p 0 µ(|ex|) p ≤ A p K t (ex, p, ∞) p . ✺✹❈❍❆P❚❊| ✷✳ P|❊▲■▼■◆❆|■❊❙ ■◆ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ■◆❚❊•|❆❚■❖◆ ❚❍❊❖|( ❋✉rt❤❡r♠♦r❡ ❢♦r ❛•• s ∈ R + ✱ µ s (ex) ≤ µ s (x)✱ s❡❡ ❢♦r ❡①❛♠♣•❡ ❬✶✻(✳ ❍❡♥❝❡✱ K t (ex, p, ∞) ≤ K t (x, p, ∞)✳ ❈♦♠❜✐♥✐♥❣ t❤❡ t✇♦ ♣r❡✈✐♦✉s ✐♥❡q✉❛•✐t✐❡s✱ ✇❡ ♦❜t❛✐♥✿ ex p ≤ A p K t (ex, p, ∞) ≤ A p K t (x, p, ∞). |❡♠❛r❦ ✷✳✷✳✶✵✳ ❚❤❡ ♣r♦♦❢ ②✐❡•❞s ❛ ❜✐t ♠♦r❡ t❤❛♥ t❤❡ ♣r♦♣♦s✐t✐♦♥✳ ■♥❞❡❡❞✱ ✐t s✉✣❝❡s t♦ ❝♦♥s✐❞❡r t❤❡ s✉♣r❡♠✉♠ ♦✈❡r ♣r♦•❡❝t✐♦♥s e ❝♦♠♠✉t✐♥❣ ✇✐t❤ |x * | t♦ ♦❜t❛✐♥ t❤❡ •❡❢t ❤❛♥❞ s✐❞❡ ✐♥❡q✉❛•✐t②✳ ❚❤✐s ✇✐•• ❜❡ ♦❢ ✐♠♣♦rt❛♥❝❡ •❛t❡r ♦♥✳ ▲❡t ✉s ♠❡♥t✐♦♥ ♦♥❡ ✜♥❛• •❡♠♠❛ ❜❡❢♦r❡ ♠♦✈✐♥❣ ♦♥ t♦ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ▲❡♠♠❛ ✷✳✷✳✶✶✳ ❋♦r ❛♥② x ∈ M✱ t ∈ R ≥0 ✱ y, z ∈ M c s✉❝❤ t❤❛t y + z = x✱ t❤❡r❡ ❡①✐sts y ′ , z ′ ∈ M c s✉❝❤ t❤❛t y ′ + z ′ = x✱ y ′ ∞ ≤ 2 x ∞ ✱ z ′ ∞ ≤ 2 x ∞ ❛♥❞✿ y ′ p + t z ′ q ≤ y p + t z q . Pr♦♦❢✳ ❙✉♣♣♦s❡ t❤❛t x ✐s ♣♦s✐t✐✈❡✱ ✇❤✐❝❤ ✐s ♣♦ss✐❜•❡ ❜② ♠✉•t✐♣•②✐♥❣ ❜② ❛ ♣❛rt✐❛• ✐s♦♠❡tr②✳ ❋✐rst✱ ♥♦t❡ t❤❛t ✐❢ p, q ≥ 1✱ ✇❡ ❝❛♥ st❛rt ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❞❡❝♦♠✲ ♣♦s✐t✐♦♥ x = E x (y) + E x (z) ✇❤❡r❡ E x ✐s t❤❡ ❝♦♥❞✐t✐♦♥❛• ❡①♣❡❝t✐♦♥ ♦♥t♦ M x ✱ ❛ s❡♠✐✜♥✐t❡ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛•❣❡❜r❛ ♦❢ M ❝♦♥t❛✐♥✐♥❣ x✳ ❙✐♥❝❡ M x ✐s ❝♦♠♠✉t❛✲ t✐✈❡✱ ✇❡ ❛r❡ r❡❞✉❝❡❞ t♦ t❤❡ ❝♦♠♠✉t❛t✐✈❡ ❝❛s❡ ✇❤✐❝❤ ✐s ❞✐r❡❝t s♦ t❤❡ ❛r❣✉♠❡♥t ✐s ❝♦♠♣•❡t❡✳ ❚❤❡♥✱ s✉♣♣♦s❡ ❢♦r ❡①❛♠♣•❡ t❤❛t p < 1 ✭q < 1 ✐s s✐♠✐•❛r✮✳ ▲❡t A = x ∞ ❛♥❞ ❞❡✜♥❡ e = 1 [0,A] (|y|)✳ ❈♦♥s✐❞❡r y ′ = ye + xe ⊥ ❛♥❞ z ′ = ze✳ ❲❡ ❤❛✈❡ y ′ + z ′ = x✱ y ′ ∞ ≤ ye ∞ + xe ⊥ ∞ ≤ 2A ❛♥❞ z ′ ∞ = x -y ′ ∞ = xe -ye ∞ ≤ 2A✳ ❙✐♥❝❡ e ✐s ❛ ❝♦♥tr❛❝t✐♦♥✱ z ′ q ≤ z q ✳ ❋✐♥❛••②✱ r❡❝❛•• t❤❛t e = 1 [0,A] (|y|) s♦ ye ⊥ p p ≥ τ (e ⊥ )A p ≥ xe ⊥ p p ❛♥❞ e ❝♦♠♠✉t❡s ✇✐t❤ |y| s♦✿ y p p = τ (|y| p ) = τ (|y| p e) + τ (|y| p e ⊥ ) = τ ((|y|) p e) + τ ((|y|) p e ⊥ ) = τ (|ye| p ) + τ ( ye ⊥ p ) = ye p p + ye ⊥ p p . ❆♥❞ ✇❡ ❝❛♥ ❝♦♥❝•✉❞❡✱ ❜② t❤❡ p✲tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t②✿ y ′ p p ≤ ye p p + xe ⊥ p p ≤ ye p p + ye ⊥ p p = y p p . ❲❡ ❝♦♥❝•✉❞❡ t❤✐s s❡❝t✐♦♥ ❜② ♣r♦✈✐♥❣ ✷✳✷✳✸✳ ✷✳✷✳ ❙❖▼❊ P|❖P❊|❚■❊❙ ❖❋ ❚❍❊ K✲❋❯◆❈❚■❖◆❆▲ ✺✺ Pr♦♦❢✳ ✐✳ ❉❡✜♥❡ b ε = b + ε✳ ❙❡t c ε = a 1/2 b -1/2 ε ✳ ❲❡ ♦♥•② ❤❛✈❡ t♦ ❝❤❡❝❦ t❤❛t c ✐s ❛ ❝♦♥tr❛❝t✐♦♥✳ c * c = b -1/2 ε a 1/2 a 1/2 b -1/2 ε = b -1/2 ε ab -1/2 ε ≤ b -1/2 ε b ε b -1/2 ε = 1 ❍❡♥❝❡✱ c ε ✐s ❛ ❝♦♥tr❛❝t✐♦♥✳ ▲❡t c ❜❡ ❛ ✇❡❛❦ * ✲•✐♠✐t ♦❢ t❤❡ c ε ✳ c ✐s ❛ ❝♦♥tr❛❝t✐♦♥ ❛♥❞ a = cbc * ✳ ✐✐✳ ■❢ a ≤ b t❤❡♥ a 2 ≤ a 1/2 ba 1/2 ✳ ◆♦✇ ❞❡✜♥❡ u t♦ ❜❡ ❛ ♣❛rt✐❛• ✐s♦♠❡tr② ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ ♣♦•❛r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ a 1/2 b 1/2 ✐✳❡ a 1/2 b 1/2 = u a 1/2 b 1/2 ✳ ❚❤❡♥ u * b 1/2 a 1/2 u = a 1/2 b 1/2 ❛♥❞ a 1/2 ba 1/2 = ub 1/2 ab 1/2 u * ≤ ub 2 u * ✳ ✐✐✐✳ ❋♦r α ∈ [1, 2]✱ ❜② ♦♣❡r❛t♦r ❝♦♥✈❡①✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥ x → x α ✱ t❤❡ r❡s✉•t ❤♦•❞s ✇✐t❤ u = 1✳ ◆♦✇ ✇❡ ♣r♦❝❡❡❞ ❜② ✐♥❞✉❝t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t ✐✐✐✳ ✐s tr✉❡ ❢♦r ❛ α ≥ 1✱ ✇❡ ✇✐•• s❤♦✇ t❤❛t ✐t ❤♦•❞s ❢♦r α ′ = 2α✳ ❇② ❤②♣♦t❤❡s✐s✱ t❤❡r❡ ❡①✐sts ❛ ♣❛rt✐❛• ✐s♦♠❡tr② u s✉❝❤ t❤❛t✿ (a + b) α ≤ 2 α-1 u(a α + b α )u * . ❆♥❞ ❜② t❤❡ ✐✐✳ t❤❡r❡ ❡①✐sts v s✉❝❤ t❤❛t✿ (a + b) 2α ≤ 2 2α-2 v(u(a α + b α )u * ) 2 v * (a + b) α ′ ≤ 2 α ′ -2 vu(a α + b α ) 2 (vu) * ≤ 2 α ′ -1 vu(a α ′ + b α ′ )(vu) * ✇❤❡r❡ t❤❡ •❛st ✐♥❡q✉❛•✐t② ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ♦♣❡r❛t♦r ❝♦♥✈❡①✐t② ♦❢ x → x 2 ♦r ♠♦r❡ ♣r❡❝✐s❡•② t❤❡ ✐♥❡q✉❛•✐t② (c + d) 2 ≤ 2c 2 + 2d 2 ❛♣♣•✐❡❞ t♦ c = a α ❛♥❞ d = b α ✳ ✐✈✳ ❉❡✜♥❡ x = a+b ❛♥❞ t❛❦❡ t✇♦ ❝♦♥tr❛❝t✐♦♥s α ❛♥❞ β s✉❝❤ t❤❛t a 1/2 = αx 1/2 ❛♥❞ b 1/2 = βx 1/2 ✳ ❲❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t α * α + β * β = s(x)✳ ❚❤❡ ♦♣❡r❛t♦r T : M → M ❞❡✜♥❡❞ ❜② T (c) = αcα * ✐s ♣♦s✐t✐✈❡ ❛♥❞ s✉❜✉♥✐t❛•✳ ❍❡♥❝❡✱ ✇❡ ❝❛♥ ❛♣♣•② t❤❡ ❏❡♥s❡♥ ✐♥❡q✉❛•✐t② ❢♦r t❤❡ ♦♣❡r❛t♦r✲❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ t → t θ ❛♥❞ ✇❡ ♦❜t❛✐♥ T (x) θ ≥ T (x θ )✳ ❙✐♠✐•❛r•②✱ βx θ β * ≤ (βxβ * ) θ ✳ ◆♦t❡ ❛•s♦ t❤❛t✱ ❛s ✇❡ ❤❛✈❡ ✉s❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣r♦♦❢✱ t❤❡r❡ ❡①✐st ♣❛rt✐❛• ✐s♦♠❡tr✐❡s u ❛♥❞ v s✉❝❤ t❤❛t x θ/2 α * αx θ/2 = uαx θ α * u * ❛♥❞ x θ/2 β * βx θ/2 = vβx θ β * v * s✐♥❝❡ ❛❞•♦✐♥ts ❛r❡ ❛•✇❛②s ❝♦♥•✉❣❛t❡❞ ❜② ❛ ♣❛rt✐❛• ✐s♦♠❡tr②✳ ◆♦✇✱ ✇❡ ❝❛♥ ❝♦♥❝•✉❞❡ ❜② t❤❡ ❢♦••♦✇✐♥❣ ❝♦♠♣✉t❛t✐♦♥✿ (a + b) θ = x θ = x θ/2 (α * α + β * β)x θ/2 = x θ/2 α * αx θ/2 + x θ/2 β * βx θ/2 = uαx θ α * u * + vβx θ β * v * ≤ u(αxα * ) θ u * + v(βxβ * ) θ v * =≤ ua θ u * + vb θ v * . ✺✻❈❍❆P❚❊| ✷✳ P|❊▲■▼■◆❆|■❊❙ ■◆ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ■◆❚❊•|❆❚■❖◆ ❚❍❊❖|( ✷✳✸ ❯•tr❛♣r♦❞✉❝ts ❙✉♣♣♦s❡ ✐♥ t❤✐s s❡❝t✐♦♥ t❤❛t M ✐s ✜♥✐t❡✳ ❲❡ ✇✐•• ❞❡♥♦t❡ ❜② M ω t❤❡ ✉•tr❛♣r♦❞✉❝t ♦❢ N ❝♦♣✐❡s ♦❢ M✳ ❍❡r❡✱ ω ✐s ❛♥ ✉•tr❛✜•t❡r ♦♥ N✳ ❙❡❡ ❬✹✹( ❢♦r t❤❡ ❞❡t❛✐•❡❞ ❝♦♥str✉❝t✐♦♥ ♦❢ tr❛❝✐❛• ✉•tr❛♣r♦❞✉❝ts✱ ✇❡ ♦♥•② r❡❝❛•• t❤❡ ❞❡✜♥✐t✐♦♥ ❤❡r❡✳ ▲❡t B = ℓ ∞ (M) ❛♥❞ I = {x ∈ B : lim ω τ (x n ) = 0}✳ I ✐s ❛ ❝•♦s❡❞ ✐❞❡❛• ♦❢ B ❛♥❞ ✇❡ ❞❡✜♥❡ M ω = B/I ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ tr❛❝❡ τ ω (x + I) = lim ω τ (x n )✳ (M ω , τ ω ) ✐s ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ ❛♥❞ (M, τ ) ❝❛♥ ❜❡ ✐s♦♠❡tr✐❝❛••② ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ s✉❜❛•❣❡❜r❛ ♦❢ ❝•❛ss❡s ♦❢ ❝♦♥st❛♥t s❡q✉❡♥❝❡s✳ ■♥ t❤❡ ♥❡①t ❝❤❛♣t❡r✱ ✇❡ ✇✐•• ♥❡❡❞ t❤❡ ❢♦••♦✇✐♥❣ •❡♠♠❛ t♦ ♠❛♥✐♣✉•❛t❡ t❤❡ K✲❢✉♥❝t✐♦♥❛• ✐♥ ❛♥ ✉•tr❛♣r♦❞✉❝t✳ ▲❡♠♠❛ ✷✳✸✳✶✳ ▲❡t x = (x n ) n∈N ∈ M ω ✱ p, q ∈ (0, ∞] ✇✐t❤ p = q✱ t❤❡♥ K(x n , L p (M), L q (M)) ❝♦♥✈❡r❣❡s ♣♦✐♥t✇✐s❡ t♦ K(x, L p (M ω ), L q (M ω )) ❛•♦♥❣ ω✳ ❋✉rt❤❡r♠♦r❡ ✐❢ p = ∞✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐s ✉♥✐❢♦r♠✳
Pr♦♦❢✳ ❲❡ tr❡❛t t❤❡ ❝❛s❡ p < q✱ t❤❡ ❝❛s❡ q > p ✐s s✐♠✐•❛r✳ ❈❤♦♦s❡ ❛ ❞❡❝♦♠♣♦s✐✲ t✐♦♥ x = y + z s✉❝❤ t❤❛t y p + t z q < K t (x, p, q) + ε✳ ❚❛❦❡ ❛ r❡♣r❡s❡♥t❛t✐✈❡

(z n ) n≥0 ♦❢ z s✉❝❤ t❤❛t lim n→∞ z n ∞ = z ∞ ✳ ❲❡ t❛❦❡ t❤✐s ♣r❡❝❛✉t✐♦♥ ✐♥ ❝❛s❡ q = ∞✳ ❲❡ ♥♦✇ ❤❛✈❡ ❛ ♥❛t✉r❛• r❡♣r❡s❡♥t❛t✐✈❡ ❢♦r y ✇❤✐❝❤ ✐s (y n ) n∈N ✇✐t❤ y n = x n -z n ✳ ❋✉rt❤❡r♠♦r❡✱ ❜② ❞❡✜♥✐t✐♦♥✿ y p = τ (|y| p ) 1/p = lim ω τ (|y n | p ) 1/p = lim ω y n p . ❙✐♥❝❡ t❤❡ s❛♠❡ ✐s tr✉❡ ❢♦r z ❛♥❞ t❤❡ q✲♥♦r♠ ✭✇✐t❤ t❤❡ ❡①❡♣t✐♦♥ ♦❢ q = ∞ ✇❤✐❝❤ ❤❛s ❜❡❡♥ ❞❡❛•t ✇✐t❤ ❛•r❡❛❞②✮✱ t❤✐s ♠❡❛♥s t❤❛t t❤❡r❡ ❡①✐sts A 1 ∈ ω s✉❝❤ t❤❛t ❢♦r ❛•• n ∈ A 1 ✱ y p + t z q -y n p -t z n q ≤ ε.
❚❤✐s ✐♠♣•✐❡s t❤❛t✿

K t (x n , p, q) ≤ K t (x, p, q) + 2ε. |❡❝✐♣r♦❝❛••②✱ ❢♦r ❛•• n ∈ N t❛❦❡ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ x n = y n + z n ∈ M s✉❝❤ t❤❛t y n p + t z n q < K t (x n , p, q) + ε✳ ❇② ❝♦♥str✉❝t✐♦♥✱ t❤❡r❡ ❡①✐sts ❛ ❝❡rt❛✐♥ ❝♦♥st❛♥t M s✉❝❤ t❤❛t ❢♦r ❛•• n ∈ N✱ x n ∞ ≤ M ✳ ❇② •❡♠♠❛ ✷✳✷✳✶✶✱ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t ❢♦r ❛•• n ∈ N✱ y n ∞ ≤ 2M ❛♥❞ z n ∞ ≤ 2M ✳ ❙♦ (y n ) n≥0 ❛♥❞ (z n ) n≥0 ❞❡✜♥❡ ❡•❡♠❡♥ts y ❛♥❞ z ∈ M ω ✳ ❲❡ ❤❛✈❡ x = y + z✱ ❛♥❞ ✇❡ ❝❛♥ ✜♥❞ A 2 ∈ ω s✉❝❤ t❤❛t ❢♦r ❛♥② n ∈ A 2 ✿ y p + t z q -y n p -t z n q ≤ ε s✐♥❝❡ ❡✈❡♥ ❢♦r q = ∞✱ z ∞ ≤ lim ω z n ∞ . ❍❡♥❝❡✱ K t (x, p, q) ≤ K t (x n , p, q) + 2ε. ❲❤✐❝❤ ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢ ♦❢ ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡✳ ■❢ p = ∞✱ ❡①t❡♥❞ K(x n , p, q) ❛t ∞ ❜② •❡tt✐♥❣ K ∞ (x n , p, q) = x n p ✳ ❚❤❡ s❛♠❡ ❝❛♥ ❜❡ ❞♦♥❡ ❢♦r K(x, p, q)✳ ❙✐♥❝❡ p = ∞✱ K(x n , p, q) ❝♦♥✈❡r❣❡s t♦ K(x, p, q) ✺✽❈❍❆P❚❊| ✷✳ P|❊▲■▼■◆❆|■❊❙ ■◆ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ■◆❚❊•|❆❚■❖◆ ❚❍❊❖|( ✉s✐♥❣ ♣r♦♣♦s✐t✐♦♥ ✷✳✹✳✷✱ K t (αu -uα, p, q) θ (b -i) -1 2θ ∞ K t ((b -i)α(b + i) -(b + i)α(b -i), p, q) θ K t (αb -bα, p, q) θ . ❲❡ ❝❛♥ ♥♦✇ ❝♦♥❝•✉❞❡ t❤❡ ♠❛✐♥ ♣r♦♦❢ ✉s✐♥❣ ❛ ♠❛tr✐① tr✐❝❦✳ Pr♦♦❢ ♦❢ •❡♠♠❛ ✷✳✹✳✶✳ ▲❡t b ∈ M ❛♥❞ α, β ∈ M + ✳ ❈♦♥s✐❞❡r✿ b ′ = 0 b b * 0 , α ′ = α 0 0 -β . ◆♦t❡ t❤❛t ✐♥ ❣❡♥❡r❛•✱ t❤❡ ♠❛tr✐① b ′ ❤❛s t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥ ❛s b ⊕ b✳ ❍❡♥❝❡✱ ❢♦r ❛•• t > 0✱ K t (b ′ , p, q) ≈ K t (b, p, q). ■♥❞❡❡❞✱ x → K t (x, p, q) ✐s ❛ q✉❛s✐✲♥♦r♠ s♦ K t (b, p, q) K t (b ′ , p, q) ❛♥❞ µ(b ′ ) = µ(b ⊕ b) ≥ µ(b) s♦ K t (b ′ , p, q) ≥ K t (b, p, q)✳ ■t ✐s ♥♦✇ str❛✐❣❤t❢♦r✇❛r❞ t♦ ❝❤❡❝❦ t❤❛t ❢♦r ❛•• t > 0✱ K t (αb + bβ, p, q) ≈ K t (α ′ b ′ -b ′ α ′ , p, q) ❛♥❞ t❤❛t ❢♦r f : x → s❣♥(x) |x| θ ✱ K t (f (α)b + bf (β), p, q) ≈ K t (f (α ′ )b ′ -b ′ f (α ′ ), p, q). ❍❡♥❝❡✱ ✐t s✉✣❝❡s t♦ ❛♣♣•② •❡♠♠❛ ✷✳✹✳✸ t♦ b ′ , α ′ , θ ❛♥❞ f t♦ ❝♦♥❝•✉❞❡ t❤❡ ♣r♦♦❢✳ ❈❤❛♣t❡r ✸ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✸✳✶ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ✇♦r❦ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r ✐s ✐♥t❡♥❞❡❞ ❛s ❛ st❡♣ t♦✇❛r❞s ❝♦♠♣•❡t✐♥❣ t❤❡ st✉❞② ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ❜❡t✇❡❡♥ L p ✲s♣❛❝❡s✳ ◆♦ s❛t✐s❢②✐♥❣ r❡s✉•ts ✇❡r❡ ❦♥♦✇♥ ✐♥ L 1,∞ ❛♥❞ L 2,∞ ✐♥ s♣✐t❡ ♦❢ t❤❡ ❡①t❡♥s✐✈❡ •✐t❡r❛t✉r❡ ♦♥ t❤❡ s✉❜•❡❝t ✇❤✐❝❤ ✐♥❝•✉❞❡s s♦♠❡ ✈❛r✐❛♥ts ✐♥ ❣❡♥❡r❛• s②♠♠❡tr✐❝ s♣❛❝❡s✳ ❚❤❡ r❡♠❛r❦❛❜•❡ ❣r♦✇t❤ ♦❢ t❤✐s t♦♣✐❝ ✐♥ t❤❡ •❛st ❞❡❝❛❞❡s ✐s t♦ ❜❡ ❛ttr✐❜✉t❡❞ t♦ t❤❡ ❝❡♥tr❛• r♦•❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ♣•❛② ✐♥ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❛♥❛•②s✐s✳ ❙✐♠✐•❛r•② t♦ t❤❡✐r ❝•❛ss✐❝❛• ❝♦✉♥t❡r♣❛rts✱ t❤❡② ❛♣♣❡❛r ❝♦♥st❛♥t•② ✇❤❡♥ t❤❡ ♥♦r♠ ♦❢ ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛• s❡q✉❡♥❝❡ ❤❛s t♦ ❜❡ ❡st✐♠❛t❡❞✱ ❛♥❞ t❤❡② ❛••♦✇ ✉s t♦ ❞❡s❝r✐❜❡ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ str✉❝t✉r❡ ♦❢ t❤❡ s♣❛♥ ♦❢ ✐♥❞❡♣❡♥❞❡♥t ♦r ❢r❡❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡s✳ ❚❤❡② ✇❡r❡ ❛ st❡♣♣✐♥❣ st♦♥❡ t♦ ❞❡✈❡•♦♣ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❛rt✐♥❣❛•❡ ✐♥❡q✉❛•✐t✐❡s ✇❤✐❝❤ ❛r❡ ❡ss❡♥t✐❛• ❛♥❞ ♣♦✇❡r❢✉• t♦♦•s t♦ tr❛♥s•❛t❡ ❝•❛ss✐❝❛• ♥♦t✐♦♥s t♦ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s❡tt✐♥❣✱ ❛s ✐••✉str❛t❡❞ ❜② t❤❡ •❛st ❝❤❛♣t❡r ♦❢ t❤✐s t❤❡s✐s✳ ❚❤❡ s❡♠✐♥❛• r❡s✉•t ♦❢ ▲✉st✲P✐q✉❛r❞ ❬✸✻( ❛♥❞ t❤❡♥ ▲✉st✲P✐q✉❛r❞ ❛♥❞ P✐s✐❡r ❬✸✼( ✇❤♦ ✜rst ❢♦♠✉•❛t❡❞ ❛♥❞ ♣r♦✈❡❞ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ t❤❡ s❡tt✐♥❣ ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ✐♥t❡❣r❛t✐♦♥ ✭❢♦r |❛❞❡♠❛❝❤❡r ✈❛r✐❛❜•❡s ❛♥❞ ✐♥ L p ✲s♣❛❝❡s✮ •❡❞ t♦ ❣❡♥❡r❛•✐s❛t✐♦♥s s♣r❡❛❞✐♥❣ ✐♥t♦ ❞✐✛❡r❡♥t ❞✐r❡❝t✐♦♥s✳ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❢r❡❡ ♣r♦❜❛❜✐•✐t② ♦r ❛♥❛•②s✐s ♦♥ t❤❡ ❢r❡❡ ❣r♦✉♣✱ t❤❡② ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ❜② P✐s✐❡r ❛♥❞ ❍❛❛❣❡r✉♣ ✐♥ ❬✷✵(✱ ❛♥❞ st✉❞✐❡❞ ❢✉rt❤❡r ✐♥ ❞✐✛❡r❡♥t ✇♦r❦s ✭s❡❡ ❢♦r ❡①❛♠♣•❡ ❬✹✷( ❛♥❞ ❬✺✺(✮✳ ❆s ♠❡♥t✐♦♥❡❞ ❜❡❢♦r❡✱ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ❛r❡ ❛•s♦ t❤❡ ♣r❡❝✉rs♦rs ♦❢ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❛rt✐♥❣❛•❡ ✐♥❡q✉❛•✐t✐❡s ✭❬✹✾(✱ ❬✷✹(✱ ❬✷✻(✮✱ ❛♥♦t❤❡r ♣✐••❛r ♦❢ t❤❡ t❤❡♦r②✱ s❡❡ ❢♦r ❡①❛♠♣•❡ ❬✷✼(✱ ❬✶✷( ❛♥❞ ❬✺✷(✳ ❈•♦s❡•② r❡•❛t❡❞ t♦ ♦✉r s✉❜•❡❝t✱ ❢♦r ♠♦r❡ t❤❛♥ ❛ ❞❡❝❛❞❡✱ ❛tt❡♥t✐♦♥ ❤❛s ❜❡❡♥ ❣✐✈❡♥ t♦✇❛r❞s ❣❡♥❡r❛• s②♠♠❡tr✐❝ s♣❛❝❡s✳ ❖♥❡ ❝❛♥ ♠❡♥t✐♦♥✱ ❢♦r ❡①❛♠♣•❡✱ t❤❡ ✇♦r❦ ♦❢ ▲✉st✲P✐q✉❛r❞ ❛♥❞ ❳✉ ✭❬✸✽(✮✱ ▲❡ ▼❡r❞② ❛♥❞ ❙✉❦♦❝❤❡✈ ✭❬✸✸(✮ ❛♥❞ ❉✐r❦s❡♥✱ ❞❡ P❛❣t❡r✱ P♦t❛♣♦✈✱ ❙✉❦♦❝❤❡✈ ✭❬✶✸(✮✳ ❇✉t ♦♥•② r❡❝❡♥t•② t❤❡ ❝❛s❡ ♦❢ q✉❛s✐✲❇❛♥❛❝❤ s♣❛❝❡s ✇❛s t❛❝❦•❡❞ ❜② P✐s✐❡r ❛♥❞ |✐❝❛r❞ ✐♥ ❬✹✽( ✇❤♦ ♣r♦✈❡❞ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ ♥♦♥❝♦♠♠✉t❛t✐✈❡ L p ✲s♣❛❝❡s ✺✾ ✻✵ ❈❍❆P❚❊| ✸✳ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ❑❍■◆❚❈❍■◆❊ ■◆❊◗❯❆▲■❚■❊❙ ❢♦r p < 1✳ ❚❤❡ •❛tt❡r ♣❛♣❡r ❣✐✈❡s t❤❡ ✜♥❛• ❦❡② ✐♥❡q✉❛•✐t② t♦ ❛♣♣•② t❤❡ ♠❡t❤♦❞ ❢♦✉♥❞ ✐♥ ❬✹✺( ❜② P✐s✐❡r✳ ❖✉r ♠❡t❤♦❞ t❛❦❡s ✐♥s♣✐r❛t✐♦♥ ❢r♦♠ ❬✶✹( ✇❤❡r❡ ❉✐r❦s❡♥ ❛♥❞ |✐❝❛r❞ ♣r♦✈❡❞ t❤❡ ✉♣♣❡r ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ ❛ ✈❡r② ❡✣❝✐❡♥t ✇❛②✳ ❲❡ ❣✐✈❡ ❛ s✐♠✐•❛r ♣r♦♦❢ ♦❢ t❤❡ •♦✇❡r ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t② ✇❤✐❝❤ ✐s ✉s✉❛••② ♦❜t❛✐♥❡❞ ❜② ❞✉❛•✐t②✳ ❚❤✐s ♣❛rt✐❛••② ❡①♣•❛✐♥s t❤❡ ❞✐✣❝✉•t② ♦❢ ♣r♦✈✐♥❣ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ L p ✲s♣❛❝❡s ❢♦r p < 1 ♦r ✐♥ ❛♥② q✉❛s✐✲❇❛♥❛❝❤ s♣❛❝❡✳ ❲❡ ♣r❡s❡♥t t✇♦ ❞✐✛❡r❡♥t r❡s✉•ts ✭❙❡❝t✐♦♥s ✸✳✷ ❛♥❞ ✸✳✸✮ ✇✐t❤ ✐♥❞❡♣❡♥❞❡♥t ♣r♦♦❢s t❤♦✉❣❤ t❤❡② ♣❛rt✐❛••② r❡•② ♦♥ t❤❡ s❛♠❡ ✐❞❡❛✳ ■♥ t❤❡ ✜rst ♦♥❡✱ ✇❡ s❤♦✇ t❤❛t t❤❡ •♦✇❡r ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t② ✐♥ L p ❢♦r p < 2 ✐♠♣•✐❡s t❤❡ •♦✇❡r ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ❢♦r ❛•• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ❜❡t✇❡❡♥ L p ❛♥❞ L ∞ ✇✐t❤ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ t❤❛t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ s♣❛❝❡✳ ❚❤✐s✱ ❝♦♠❜✐♥❡❞ ✇✐t❤ ❦♥♦✇♥ r❡s✉•ts✱ ❞✐r❡❝t•② ✐♠♣•✐❡s ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ L 1,∞ ✱ ✇❤✐❝❤ ❝♦✉•❞ ♥♦t ❜❡ r❡❛❝❤❡❞ ❜❡❢♦r❡ ❞✉❡ t♦ t❤❡ ✐♥❛♣♣•✐❝❛❜✐•✐t② ♦❢ ✐♥t❡r♣♦•❛t✐♦♥ ♦r ❞✉❛•✐t② t❡❝❤♥✐q✉❡s ✐♥ t❤✐s ❝❛s❡✳ ❆ ♠♦t✐✈❛t✐♦♥ t♦ ♣r♦✈❡ t❤✐s •❛st r❡s✉•t ✇❛s t❤❛t ✐t ❛••♦✇s ✉s t♦ ♣r♦✈❡ t❤❡ ✇❡❛❦✲1✲ ❜♦✉♥❞❡❞♥❡ss ♦❢ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦rs ✐♥ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s❡tt✐♥❣ ❢♦r ❍✐•❜❡rt✲✈❛•✉❡❞ ❦❡r♥❡•s ✭❬✸✾(✮ ❞✐r❡❝t•② ❢r♦♠ t❤❡ s❝❛•❛r✲✈❛•✉❡❞ ❦❡r♥❡• ❝❛s❡ ✭❬✹✶(✮✱ s❡❡ ❝❤❛♣t❡r ✹✳✱ ❝♦r♦••❛r② ✹✳✹✳✷✳ ❚❤❡ s❡❝♦♥❞ t❤❡♦r❡♠ ❣✐✈❡s ❛ ❞❡t❡r♠✐♥✐st✐❝ ❡q✉✐✈❛•❡♥t ❢♦r ✧❢r❡❡ ❛✈❡r❛❣❡s✧ ✐♥ ❡✈❡r② ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ❜❡t✇❡❡♥ L p ✲s♣❛❝❡s✳ ❚❤❡ r❡♠❛r❦❛❜•❡ ❢❡❛t✉r❡ ❤❡r❡ ✐s t❤❛t ✐ts ❢♦r♠✉•❛t✐♦♥ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ✇❤❡t❤❡r p < 2 ♦r p > 2✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✐t ❤♦•❞s ✐♥ L 2,∞ ✇❤✐❝❤ ✐s ❛ tr✐❝❦② ❝❛s❡ s✐♥❝❡ ♥❡✐t❤❡r ♦❢ t❤❡ t✇♦ ✉s✉❛• ❢♦r♠✉•❛s ❢♦r ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✇♦r❦ ✭s❡❡ ❙❡❝t✐♦♥ ✸✳✺✮✳ ◆♦t❡ t❤❛t ❛ ❞❡t❡r♠✐♥✐st✐❝ ❢♦r♠✉•❛ ✇❛s ❛•r❡❛❞② ❢♦✉♥❞ ✉s✐♥❣ ✐♥t❡r♣♦•❛t✐♦♥ ♠❡t❤♦❞s ❜② P✐s✐❡r ✐♥ ❬✹✻(✳ ❖✉r ❡q✉✐✈❛•❡♥t ✐s •❡ss tr❛❝t❛❜•❡ t❤❛♥ t❤❡ ✉s✉❛• ❢♦r♠✉•❛s✳ ■t ✐s ♦❜t❛✐♥❡❞ ❜② ✜rst ♣r♦✈✐♥❣ t❤❛t ❛♥② s❡q✉❡♥❝❡ ♦❢ ♦♣❡r❛t♦rs (x i ) ❛❞♠✐ts ❛ ❢❛❝t♦r✐s❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠ α(u i ) + (u i )β ✇❤❡r❡ α ❛♥❞ β ❛r❡ ♣♦s✐t✐✈❡ ♦♣❡r❛t♦rs ❛♥❞ (u i ) ❤❛s ❣♦♦❞ ♣r♦♣❡rt✐❡s✳ ❚❤❡♥✱ ✐❢ (ξ i ) ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❢r❡❡ ❍❛❛r ✉♥✐t❛r✐❡s✱ t❤❡ ♥♦r♠ ♦❢ x i ⊗ ξ i ❤❛♣♣❡♥s t♦ ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ♥♦r♠ ♦❢ α ⊕ β ✐♥ ❛•• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ❜❡t✇❡❡♥ L p ✲s♣❛❝❡s✳ ❚❤✐s ②✐❡•❞s ❛ ♥❡✇ ♣r♦♦❢ ♦❢ ❢r❡❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ❢♦r p < 1✳ ■t ❞♦❡s ♥♦t ❛♣♣•② t♦ |❛❞❡♠❛❝❤❡r ✈❛r✐❛❜•❡s s✐♥❝❡ ✐t r❡•✐❡s ♦♥ ❍❛❛❣❡r✉♣✬s ✐♥❡q✉❛•✐t② ✭❬✶✾(✮ t♦❣❡t❤❡r ✇✐t❤ ❛ ❍ö•❞❡r ✐♥❡q✉❛•✐t② ❢♦r ❛♥t✐✲❝♦♠♠✉t❛t♦rs ✭✷✳✹✳✶✮ ❢♦✉♥❞ ✐♥ ❬✺✹(✳ ▲❡t ✉s ♥♦✇ st❛t❡ t❤❡ ♠❛✐♥ r❡s✉•ts ♦❢ t❤✐s ❝❤❛♣t❡r✳ ❲❡ r❡❢❡r t♦ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✱ ❢♦r t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ ♠♦st ♥♦t❛t✐♦♥s ❛♣♣❡❛r✐♥❣ t❤❡r❡❛❢t❡r✳ ❈♦♥s✐❞❡r (A, τ A ) ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♣r♦❜❛❜✐•✐t② s♣❛❝❡ ✭τ A (1) = 1✮ ❛♥❞ ξ = (ξ i ) i∈N ❛ s❡q✉❡♥❝❡ ♦❢ ❡•❡♠❡♥ts ✐♥ A✳ ❋♦r x ✐♥ S(M)✱ ✇❡ ❞❡✜♥❡ Gx = n≥0 x n ⊗ ξ n ✐♥ M A := M⊗A✳ |❡❝❛•• t❤❛t M ✇✐•• ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ M ⊗ 1 ✐♥ M A ✳ ❋✐① E ❛ s②♠♠❡tr✐❝ s♣❛❝❡✳ ❚❤❡ q✉❛♥t✐t② ✇❡ ✇❛♥t t♦ ❡st✐♠❛t❡ ✐s x H E := Gx E(M A ) ✇❤✐❝❤ ✐s ❛ q✉❛s✐✲♥♦r♠ ♦♥ S(M)✳ ❉❡♥♦t❡ ❜② H E (M) ✭H E ✐❢ t❤❡r❡ ✐s ♥♦ ❛♠❜✐❣✉✐t②✮ t❤❡ ❝♦♠♣•❡t✐♦♥ ♦❢ S(M) ❢♦r . H E ✭❢♦r p ∈ (0, ∞]✱ ✇❡ ✇r✐t❡ H p := H Lp ✮✳ |❡♠❛r❦ ✸✳✶✳✶✳ ▲❡t p, q ∈ (0, ∞]✱ t❤❡ ❝♦✉♣•❡ (H p , H q ) ✐s ❛ ❝♦♠♣❛t✐❜•❡ ❝♦✉♣•❡ ♦❢ ❇❛♥❛❝❤ s♣❛❝❡s ✐♥ t❤❡ s❡♥s❡ ♦❢ ✐♥t❡r♣♦•❛t✐♦♥✳ ■♥❞❡❡❞✱ ❢♦r ❛•• r > 0✱ H r ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ ❛ ❝•♦s❡❞ s✉❜s♣❛❝❡ ♦❢ L r (M A ) ❜② ❡①t❡♥❞✐♥❣ t❤❡ ♠❛♣ G✳ ❲✐t❤ t❤✐s ♥♦t❛t✐♦♥s✱ t❤❡ ✜rst ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✭❬✸✻(✱❬✸✼(✮ ✸✳✶✳ ■◆❚|❖❉❯❈❚■❖◆ ✻✶ st❛t❡ t❤❛t✱ ❛ss✉♠✐♥❣ t❤❛t ξ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥❞❡♣❡♥❞❡♥t |❛❞❡♠❛❝❤❡r ✈❛r✐❛❜•❡s✱ H p = R p + C p ✐❢ p ∈ [1, 2] R p ∩ C p ✐❢ p ∈ [2, ∞) ✇✐t❤ ❡q✉✐✈❛•❡♥t ♥♦r♠s✳ ❚❤✐s ♠♦t✐✈❛t❡s t❤❡ ❢♦••♦✇✐♥❣ ❞❡✜♥✐t✐♦♥✳ ❉❡✜♥✐t✐♦♥ ✸✳✶✳✷✳ ▲❡t E ❜❡ ❛ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡✳ ❉❡✜♥❡ t❤❡ ❢♦••♦✇✐♥❣ t✇♦ ♣r♦♣❡rt✐❡s ♦❢ E✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ M✿ Kh + (E, M) ⇔ ∀x ∈ S(M), x H E ≈ x R E ∩C E , ❛♥❞✿ Kh -(E, M) ⇔ ∀x ∈ S(M), x H E ≈ x R E +C E . ❚❤❡ ♠❛✐♥ r❡s✉•ts ♦❢ s❡❝t✐♦♥s ✸✳✷ ❛♥❞ ✸✳✸ ❝♦♠❡ ❢r♦♠ t❤❡ st✉❞② ♦❢ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥s✳ ▲❡t ✉s ❣✐✈❡ t❤❡✐r ❞❡✜♥✐t✐♦♥ r✐❣❤t ❛✇❛②✳ ❉❡✜♥✐t✐♦♥ ✸✳✶✳✸✳ ❋♦r ❛♥② x ∈ S(M) ❛♥❞ p ∈ (0, 1]✳ ❉❡✜♥❡ m p (x) = inf{ Ry p p + Cz p p : x = z + y, y, z ∈ S(M)}. ❲❡ s❛② t❤❛t y, z ∈ S(M) ✐s ❛♥ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ x ✐♥ L p ✐❢ y + z = x ❛♥❞ Ry p p + Cz p p = m p (x)✳ ❚❤❡ ✐♥t✉✐t✐♦♥ ❜❡❤✐♥❞ t❤✐s ❞❡✜♥✐t✐♦♥ ✐s t❤❛t ❞✐str✐❜✉t✐♦♥s ♦❢ ♦♣t✐♠❛• ❞❡❝♦♠✲ ♣♦s✐t✐♦♥s ♦❢ x s❤♦✉•❞ s♦♠❡❤♦✇ ❛♣♣r♦❛❝❤ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ Gx✳ ❚❤♦✉❣❤ ✈❡r② ✈❛❣✉❡✱ t❤✐s ✐❞❡❛ ✐s ♣❛rt✐❛••② ❝♦♥✜r♠❡❞ ❜② t❤❡ t❤❡♦r❡♠ ✸✳✶✳✽✳ ❖✉r ✜rst r❡s✉•t ✐s ❛ ♥❡❣❛t✐✈❡ ♦♥❡✱ ✇❡ ❡①❤✐❜✐t ❝♦✉♥t❡r❡①❛♠♣•❡s t♦ ♣r♦✈❡ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✿ Pr♦♣♦s✐t✐♦♥ ✸✳✶✳✹✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ξ i ❛r❡ ❢r❡❡ ❍❛❛r ✉♥✐t❛r✐❡s ♦r |❛❞❡♠❛❝❤❡r ✈❛r✐❛❜•❡s ❛♥❞ t❤❛t M = B(ℓ 2 )✳ ❚❤❡r❡ ✐s ♥♦ ❝♦♥st❛♥t c s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ✜♥✐t❡ s❡q✉❡♥❝❡ x = (x n ) n≥0 ∈ M N ✿ i≥0 x i ⊗ ξ i 2,∞ ≤ c x R2,∞+C2,∞ . ❙✐♠✐•❛r•②✱ t❤❡r❡ ✐s ♥♦ ❝♦♥st❛♥t c s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ✜♥✐t❡ s❡q✉❡♥❝❡ x = (x n ) n≥0 ∈ M N ✿ x R2,∞∩C2,∞ ≤ c i≥0 x i ⊗ ξ i 2,∞ . ❲❡ ♣r❡s❡♥t t✇♦ ♣r♦♦❢s ♦❢ t❤✐s ♣r♦♣♦s✐t✐♦♥✳ ■♥ t❤❡ ✜rst✱ ✇❡ ❝♦♥str✉❝t ❡①♣•✐❝✐t ❝♦✉♥t❡r✲❡①❛♠♣•❡s✳ Pr♦✈✐♥❣ t❤❡ r❡q✉✐r❡❞ ❡st✐♠❛t❡ ✐♥ t❤✐s ❝❛s❡ ✐s ♥♦t ✐♠♠❡❞✐❛t❡ ❛♥❞ r❡•✐❡s ✐♥ ♣❛t✐❝✉•❛r ♦♥ ❬✶✹(✳ ■♥ t❤❡ s❡❝♦♥❞✱ ✇❡ ❝r✉❝✐❛••② ✉s❡ t❤❡ ❙❝❤✉r✲❍♦r♥ •❡♠♠❛ ❢♦r t❤❡ ❝♦♥str✉❝t✐♦♥✳ ❚❤❡ ♠❡t❤♦❞ ✇♦r❦s ✐♥ ❛ ♠✉❝❤ ♠♦r❡ ❣❡♥❡r❛• s❡tt✐♥❣ ❛♥❞ ❝♦♠❜✐♥❡❞ ✇✐t❤ t❤❡♦r❡♠ ✸✳✶✳✽✱ ❡♥❛❜•❡s ✉s t♦ ♣r♦✈❡ t❤❡ ❢♦••♦✇✐♥❣✿ ✻✷ ❈❍❆P❚❊| ✸✳ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ❑❍■◆❚❈❍■◆❊ ■◆❊◗❯❆▲■❚■❊❙ ❚❤❡♦r❡♠ ✸✳✶✳✺✳ ▲❡t E ❜❡ ❛ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡ ✇✐t❤ t❤❡ ❋❛t♦✉ ♣r♦♣❡rt②✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ξ i ❛r❡ ❢r❡❡ ❍❛❛r ✉♥✐t❛r✐❡s ❛♥❞ M = B(ℓ 2 )⊗L ∞ (0, 1)✳ ❚❤❡♥✱ Kh + (E, M) ⇔ E ✐s r✐❣❤t✲2✲♠♦♥♦t♦♥❡ ❛♥❞ ✐❢ ❢✉rt❤❡r♠♦r❡ t❤❡r❡ ❡①✐sts p > 0 s✉❝❤ t❤❛t E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ L p ❛♥❞ L ∞ ✱ Kh -(E, M) ⇔ E ✐s •❡❢t✲2✲♠♦♥♦t♦♥❡. ❚❤❡ ♥❡①t t❤❡♦r❡♠ ❤❛s t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ r❡q✉✐r✐♥❣ ✈❡r② •✐tt•❡ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ✈❛r✐❛❜•❡s ❝♦♥s✐❞❡r❡❞✳ ❲❡ ♣r♦✈❡ t❤❛t ✐❢ t❤❡ •♦✇❡r ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t② ❤♦•❞s ❢♦r s♦♠❡ p✱ ✐t ❛•s♦ ❤♦•❞s ❢♦r q > p ✇✐t❤ s❛♠❡ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♦r❡♠ ✸✳✶✳✻✳ ▲❡t p ∈ (0, 1]✳ ❙✉♣♣♦s❡ t❤❛t✿ • t❤❡ ξ i ❛r❡ ♦rt❤♦♥♦r♠❛• ✐♥ L 2 (A)✱ • t❤❡ ξ i ✈❡r✐❢② t❤❡ •♦✇❡r ❜♦✉♥❞ ♦❢ t❤❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t② ✐♥ L p ✐✳❡ Kh -(L p , M) ❤♦•❞s✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ y, z s✉❝❤ t❤❛t x = y + z ❛♥❞ ❢♦r ❛•• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ E ❜❡t✇❡❡♥ L p ❛♥❞ L ∞ ✿ i≥0 y i y * i 1/2 E(M) + i≥0 z * i z i 1/2 E(M) c i≥0 x i ⊗ξ i E(M A ) . ❚❤❡ ❝♦♥st❛♥t c ♦♥•② ❞❡♣❡♥❞s ♦♥ A ❛♥❞ p✳ ❚❤❡ ♣r♦♦❢ ✐s ❛ ❞✉❛• ✈❡rs✐♦♥ ♦❢ t❤❡ ❛r❣✉♠❡♥t ✉s❡❞ ✐♥ ❬✶✹(✳ ❲❡ ✉s❡ t❤❡ ❑❤✐♥t✲ ❝❤✐♥❡ ✐♥❡q✉❛•✐t② ✐♥ L p ♥♦t ♦♥•② ❢♦r ❛♥ ❡•❡♠❡♥t x ∈ S(M) ❜✉t ❛•s♦ ❢♦r ❡•❡♠❡♥ts ♦❢ t❤❡ ❢♦r♠ ex ✇❤❡r❡ e ✐s ❛ ✇❡•• ❝❤♦s❡♥ ♣r♦•❡❝t✐♦♥ ✐♥ M✳ ❚❤✐s ❡♥❛❜•❡s ✉s t♦ ♦❜t❛✐♥ ❛ ❝♦♥tr♦• ✐♥ t❡r♠s ♦❢ K✲❢✉♥❝t✐♦♥❛•s ✇❤✐❝❤✱ ❜② ♣r♦♣♦s✐t✐♦♥ ✶✳✸✳✺✱ ✐♠♠❡❞✐❛t❡•② ✐♠✲ ♣•✐❡s t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡ ❛s ❛ ❝♦r♦••❛r②✳ ❋♦r t❤✐s ✐❞❡❛ t♦ ✇♦r❦ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ y, z ❤❛s t♦ ❜❡ ❝•♦s❡ ❡♥♦✉❣❤ t♦ ❛♥ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥✳ |❡♠❛r❦ ✸✳✶✳✼✳ ❚❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ ♦♥ ξ ✐s ❛❝t✉❛••② ✐♥❞❡♣❡♥❞❡♥t ♦❢ p ❢♦r p < 2✳ ❚❤✐s ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❬✹✽( ✭t♦ ❣♦ ❢r♦♠ p t♦ q < p✮ ❛♥❞ t❤❡♦r❡♠ ✸✳✷✳✶ ✭t♦ ❣♦ ❢r♦♠ p t♦ q > p✮✳ ❚❤✐s ♠❡❛♥s t❤❛t ❛♣♣•②✐♥❣ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡ ✇❡ ❝❛♥ ♣r♦✈❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ L 1,∞ ❢♦r ❛♥② s❡q✉❡♥❝❡ ♦❢ ✈❛r✐❛❜•❡s ξ i t❤❛t ✈❡r✐✜❡s ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ L p ❢♦r s♦♠❡ p < 2✳ ▼♦r❡ ❞❡t❛✐•s ❛r❡ ❣✐✈❡♥ ✐♥ ❈♦r♦••❛r② ✸✳✷✳✷✳ ◆♦t❡ ❛•s♦ t❤❛t ❜② r❡♠❛r❦ ✸✳✷✳✾✱ t❤❡♦r❡♠ ✸✳✶✳✻ ❛•s♦ ❤♦•❞s ❢♦r p ∈ (1, 2)✳ ❚❤❡ s❡❝♦♥❞ t❤❡♦r❡♠ s✉♠♠❛r✐③❡s t❤❡ r❡s✉•ts ❢♦✉♥❞ ✐♥ s❡❝t✐♦♥ 3.3✳ ❇② ♣✉s❤✐♥❣ ❢✉rt❤❡r t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥s ✐♥ L 1 ❛♥❞ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❦❡② ✐♥❡q✉❛•✐t② ❞✉❡ t♦ |✐❝❛r❞ ✭✷✳✹✳✶✮ ✇❡ ♦❜t❛✐♥ ❛ ♥❡✇ ♣r♦♦❢ ♦❢ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ ❛•• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ❜❡t✇❡❡♥ L p ✲s♣❛❝❡s ✐❢ t❤❡ ✈❛r✐❛❜•❡s ❛r❡ ❢♦r ❡①❛♠♣•❡ ❢r❡❡ ✉♥✐t❛r✐❡s✳ ❲❡ ❛•s♦ ❤❛✈❡ ❛ ✧❞❡t❡r♠✐♥✐st✐❝✧ ❡q✉✐✈❛•❡♥t ♦❢ i≥0 x i ⊗ξ i 2,∞ ✇❤✐❝❤ ✐s ❤♦✇❡✈❡r •❡ss ❡①♣•✐❝✐t t❤❛♥ t❤❡ ✉s✉❛• ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s✳ ❚❤❡♦r❡♠ ✸✳✶✳✽✳ ▲❡t x = (x i ) i≥0 ∈ S(M)✳ ❚❤❡r❡ ❡①✐st α, β ∈ M + c ❛♥❞ u = (u i ) i≥0 ∈ S(M) s✉❝❤ t❤❛t✿ • s(α) ≤ i≥0 u i u * i ≤ 1✱ • s(β) ≤ i≥0 u * i u i ≤ 1✱ • ❢♦r ❛•• i ≥ 0✱ x i = u i β + αu i ✳ ❋✉rt❤❡r♠♦r❡✱ s✉♣♣♦s❡ t❤❛t✿ • t❤❡ ξ i ❛r❡ ♦rt❤♦♥♦r♠❛• ✐♥ L 2 (A) • t❤❡ ξ i ✈❡r✐❢② t❤❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t② ✐♥ L ∞ ✐✳❡ Kh + (L ∞ , E) ❤♦•❞s✳ ❚❤❡♥ ❢♦r ❛•• p ∈ (0, ∞) ❛♥❞ E ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ L p ❛♥❞ L ∞ ✿ α E(M) + β E(M) ≈ c i≥0 x i ⊗ ξ i E(M A ) , ✇❤❡r❡ c ♦♥•② ❞❡♣❡♥❞s ♦♥ p ❛♥❞ t❤❡ ❝♦♥st❛♥t c ∞ ❛♣♣❡❛r✐♥❣ ✐♥ ✭❄❄✮✳ ❆♣❛rt ❢r♦♠ t❤❡ ✐♥❡q✉❛•✐t② ❢♦✉♥❞ ✐♥ ❬✺✹(✱ t❤❡ ♣r♦♦❢ ✐s ❜❛s❡❞ ♦♥ ❛ s❤♦rt ❞✉❛•✐t② ❛r❣✉♠❡♥t ❛♥❞ t❤❡ ❢❛❝t t❤❛t ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥s ❛•✇❛②s ❡①✐st ✐♥ L 1 ✳ ✸✳✷ ❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥s ✐♥ L p ❢♦r p < 2 ✸✳✷✳✶ ▼❛✐♥ r❡s✉•t ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ■♥ t❤✐s s❡❝t✐♦♥✱ t❤❡ ✈❛r✐❛❜•❡s ξ i ✇✐•• ❛•✇❛②s s❛t✐s❢② t❤❡ ❢♦••♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ ✶✳ t❤❡ ξ i ❛r❡ ♦rt❤♦♥♦r♠❛• ✐♥ L 2 (A)✱ ✷✳ t❤❡ ξ i ✈❡r✐❢② t❤❡ •♦✇❡r ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t② ❢♦r s♦♠❡ p ≤ 1✳ ▼♦r❡ ♣r❡✲ ❝✐s❡•②✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t B p s✉❝❤ t❤❛t ❢♦r ❛•• x ∈ S(M)✱ m p (x) 1/p ≤ B p Gx p . ✭✸✳✶✮ ❚②♣✐❝❛• ❡①❛♠♣•❡s ♦❢ s✉❝❤ ✈❛r✐❛❜•❡s ✐♥❝•✉❞❡ ❢r❡❡ ❍❛❛r ✉♥✐t❛r✐❡s ♦r |❛❞❡♠❛❝❤❡r ✈❛r✐❛❜•❡s✳ ❲❡ ❢♦❝✉s ♦♥ •♦✇❡r ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐✳❡ ♦❢ t❤❡ t②♣❡✿ x R E +C E Gx E , ❢♦r E ❛ s②♠♠❡tr✐❝ s♣❛❝❡ ❛♥❞ x ∈ S(M)✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐♥❡q✉❛•✐t② ♣r❡s❡♥ts ♥♦ ❞✐✣❝✉•t② ✐♥ t❤❡ ♠♦t✐✈❛t✐♥❣ ❡①❛♠♣•❡ ♦❢ L 1,∞ ❛s ✇❡ ✇✐•• s❡❡ •❛t❡r✳ ■♥ ❬✶✹(✱ ✐t ✐s s❤♦✇♥ t❤❛t ❜② ❛♣♣•②✐♥❣ ♠✉•t✐♣•❡ t✐♠❡s ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t② ✐♥ L ∞ ❢♦r ❛♥ ❡•❡♠❡♥t x✱ ♦♥❡ ❝❛♥ ♦❜t❛✐♥ ❛ ♠❛•♦r❛t✐♦♥ ♦❢ µ(Gx)✳ ❚❤♦✉❣❤ ✐t ✐s •❡ss ❞✐r❡❝t✱ ♦✉r ♠❡t❤♦❞ ✐s s✐♠✐•❛r ❛♥❞ ❜② ✉s✐♥❣ t❤❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t② ✐♥ L p ❢♦r p < 2 ✇❡ ♦❜t❛✐♥ ❛ ♠✐♥♦r❛t✐♦♥ ♦❢ t❤❡ K✲❢✉♥❝t✐♦♥❛• ♦❢ Gx✳ ❚❤❡ ♠❛✐♥ t❤❡♦r❡♠ ♦❢ t❤✐s s❡❝t✐♦♥ ✐s t❤❡ ❢♦••♦✇✐♥❣✳ ✻✹ ❈❍❆P❚❊| ✸✳ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ❑❍■◆❚❈❍■◆❊ ■◆❊◗❯❆▲■❚■❊❙ ❚❤❡♦r❡♠ ✸✳✷✳✶✳ ▲❡t p ∈ (0, 1] s✉❝❤ t❤❛t ✭✸✳✶✮ ❤♦•❞s✳ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C p s✉❝❤ t❤❛t ❢♦r ❛•• x ∈ S(M) t❤❡r❡ ❡①✐st y, z ∈ S(M) s✉❝❤ t❤❛t y + z = x ❛♥❞ ❢♦r ❛•• t ≥ 0✱ K t (Ry, p, ∞) ≤ C p K t (Gx, p, ∞) ❛♥❞ K t (Cz, p, ∞) ≤ C p K t (Gx, p, ∞)✳ ❚❤❡ ❝♦♥st❛♥t C p ♦♥•② ❞❡♣❡♥❞s ♦♥ A p ❛♥❞ B p ✇❤✐❝❤ ❛♣♣❡❛r ✐♥ ✭✶✳✷✮ ❛♥❞ ✭✸✳✶✮ r❡s♣❡❝t✐✈❡•②✳ ▲❡t ✉s ❤✐❣❤•✐❣❤t s♦♠❡ ❝♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡ t❤❡♦r❡♠✳ ❙✐♥❝❡ ✇❡ ❤❛✈❡ ❛ ❝♦♥tr♦• ♦♥ t❤❡ K✲❢✉♥❝t✐♦♥❛•✱ ✐t ❡①t❡♥❞s t♦ ❛•• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡s ❜❡t✇❡❡♥ L p ✲s♣❛❝❡s ❜② t❤❡♦r❡♠ ✶✳✸✳✺ ❛♥❞ t❤✉s ✇❡ ♦❜t❛✐♥ t❤❡♦r❡♠ ✸✳✶✳✻ ♠❡♥t✐♦♥❡❞ ✐♥ t❤❡ ♣r❡•✐♠✐♥❛r✐❡s ❛s ❛ ❝♦r♦••❛r②✳ ■t ❛•s♦ ❛••♦✇s ✉s t♦ ♣r♦✈❡ t❤❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ L 1,∞ ✳ ❈♦r♦••❛r② ✸✳✷✳✷ ✭❚❤❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t② ✐♥ L 1,∞ ✮✳ ❋♦r ❛♥② x ∈ S(M)✱ Gx 1,∞ ≈ x R1,∞+C1,∞ . Pr♦♦❢✳ ❇② r❡♠❛r❦ ✸✳✶✳✼✱ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t p < 1✳ ❚❤❡ ✐♥❡q✉❛•✐t②✿ inf{ Ry 1,∞ + Cz 1,∞ : x = y + z, y, z ∈ S(M)} Gx 1,∞ ✐s ❣✐✈❡♥ ❜② t❤❡♦r❡♠ ✸✳✶✳✻ ❛♣♣•✐❡❞ t♦ E = L 1,∞ ✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐♥❡q✉❛•✐t② ✐s ❝•❛ss✐❝❛•✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ♠❛♣ Ry → Gy ✐s ❛ ❝♦♥tr❛❝t✐♦♥ ♦♥ L p ❛♥❞ L 2 ✱ ❤❡♥❝❡✱ ❜② ✐♥t❡r♣♦•❛t✐♦♥✱ ✐t ✐s ❜♦✉♥❞❡❞ ♦♥ L 1,∞ ✳ ❇② ❛❞•✉♥❝t✐♦♥✱ t❤❡ s❛♠❡ ✐s tr✉❡ ❢♦r Cz → Gz✳ ❍❡♥❝❡✿ Gx 1,∞ Gy 1,∞ + Gz 1,∞ Ry 1,∞ + Cz 1,∞ . |❡♠❛r❦ ✸✳✷✳✸✳ ❲❡ ❡♠♣❤❛s✐③❡❞ t❤❡ ♣r❡✈✐♦✉s ❝♦r♦••❛r② ❜❡❝❛✉s❡ ✐t ❝♦♥❝❡r♥s ❛ s♣❡❝✐❛• ❝❛s❡ t❤❛t ♠♦t✐✈❛t❡❞ ♦✉r ✇♦r❦ ❜✉t t❤❡ ❡①❛❝t s❛♠❡ ♣r♦♦❢ ✇♦r❦s ❢♦r ❛ ❣❡♥❡r❛• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ L p ✱ p < 2 ❛♥❞ L 2 ✉s✐♥❣ ✐♥t❡r♣♦•❛t✐♦♥ ❢♦r t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❛♥❞ t❤❡♦r❡♠ ✸✳✶✳✻ ❢♦r t❤❡ •♦✇❡r ❜♦✉♥❞✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ •❡t p < 2 ❛♥❞ E ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ p ❛♥❞ 2 t❤❡♥✿ H E = R E + C E ✇✐t❤ ❡q✉✐✈❛•❡♥t q✉❛s✐✲♥♦r♠s✳ |❡♠❛r❦ ✸✳✷✳✹✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ♣r❡❝❡❞✐♥❣ r❡♠❛r❦ ❛♥❞ t❤❡♦r❡♠ ✸✳✷✳✶✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦••♦✇✐♥❣✳ ▲❡t p, q ∈ (0, 2]✱ p ≤ q✳ ❚❤❡♥ ❢♦r ❛•• x ∈ S(M)✱ K t (x, H p , H q ) K t (Gx, p, q). ❯s✐♥❣ t❤❡ t❡r♠✐♥♦•♦❣② ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✹✸(✱ t❤❡ ❝♦✉♣•❡ (H p , H q ) ✐s K✲❝•♦s❡❞ ✐♥ (L p (M A ), L q (M A )) ✭r❡❝❛•• t❤❛t H p ❛♥❞ H q ❛r❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ s✉❜s♣❛❝❡s ♦❢ L p (M A ) ❛♥❞ L q (M A ) ❜② r❡♠❛r❦ ✸✳✶✳✶✮✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ H p ✲s♣❛❝❡s ❜❡❤❛✈❡ ✇❡•• ✇✐t❤ r❡s♣❡❝t t♦ ✐♥t❡r♣♦•❛t✐♦♥ ❢♦r p ≤ 2✳ ✸✳✷✳ ❙❖▼❊ P|❖P❊|❚■❊❙ ❖❋ ❖P❚■▼❆▲ ❉❊❈❖▼P❖❙■❚■❖◆❙ ■◆ L P ❋❖| P < 2✻✺ Pr♦♦❢✳ ▲❡t t > 0 ❛♥❞ ♥♦t❡ t❤❛t L p + tL q ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ L p ❛♥❞ L q ✳ ❯s✐♥❣ t❤❡ ❢❛❝t t❤❛t L q ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ L p ❛♥❞ L ∞ ❛♥❞ ❜② ❛♣♣•②✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡✱ t❤✐s ♠❡❛♥s t❤❛t L p + tL q ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ L p ❛♥❞ L ∞ ✳ ▲❡t x ∈ S(M)✳ ❚❤✐s ♠❡❛♥s ❜② t❤❡♦r❡♠ ✸✳✷✳✶ t❤❛t t❤❡r❡ ❡①✐sts y, z ∈ S(M) s✉❝❤ t❤❛t x = y + z ❛♥❞✿ K t (Ry, p, q) + K t (Cz, p, q) K t (Gx, p, q).
❍❡♥❝❡✱ ❜② r❡♠❛r❦ ✷✳✷✳✽✱ ✇❡ ❤❛✈❡✿

K t (y, R p , R q ) + K t (z, C p , C q ) K t (Gx, p, q). ▲❡t y 1 , y 2 , z 1 , z 2 ∈ S(M) s✉❝❤ t❤❛t y = y 1 + y 2 ✱ z = z 1 + z 2 ✱ y 1 Rp + t y 2 Rq K t (y, R p , R q ) ❛♥❞ z 1 Rp + t z 2 Rq K t (z, C p , C q ) ▲❡t x 1 = y 1 + z 1 ❛♥❞ x 2 = y 2 + z 2 ✱ ❝♦♠❜✐♥✐♥❣ t❤❡ ♣r❡✈✐♦✉s ✐♥❡q✉❛•✐t✐❡s✿ x 1 Rp+Cp + t x 2 Cp+Cq K t (Gx, p, q). ❙✐♥❝❡✱ ❜② t❤❡ ♣r❡✈✐♦✉s r❡♠❛r❦ R p + C p ≈ H p ❛♥❞ R q + C q ≈ H q ✱ ✇❡ ♦❜t❛✐♥✿ K t (x, H p , H q ) K t (Gx, p, q). ✸✳✷✳✷ ❋✐rst st❡♣s t♦✇❛r❞s t❤❡ ♣r♦♦❢ ■♥ t❤✐s ♣❛rt✱ ✇❡ ♣r❡s❡♥t t❤❡ ♠❛✐♥ ✐❞❡❛s t❤❛t ✇✐•• ❛••♦✇ ✉s t♦ ♣r♦✈❡ t❤❡♦r❡♠ ✸✳✷✳✶✳ ❚❤❡ ❝❡♥tr❛• ♦♥❡ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ♣r♦♣♦s✐t✐♦♥ ✸✳✷✳✺✳ ❙t❛rt✐♥❣ ✇✐t❤ ❛♥ ❡•❡♠❡♥t x✱ ✇❡ ✉s❡ t❤❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t② ♦♥ e(Gx) ❢♦r ✇❡•• ❝❤♦s❡♥ ♣r♦•❡❝t✐♦♥s e ∈ P(M) ❛♥❞ t❤❛♥❦s t♦ ♣r♦♣♦s✐t✐♦♥ ✷✳✷✳✾ ✇❡ ❞❡❞✉❝❡ t❤❡ ❡①♣❡❝t❡❞ ❝♦♥tr♦• ♦♥ K✲ ❢✉♥❝t✐♦♥❛•s✳ ❚❤❡r❡ ❛r❡✱ ❤♦✇❡✈❡r✱ t✇♦ t❡❝❤♥✐❝❛• ❞✐✣❝✉•t✐❡s✳ ❚❤❡ ✜rst ♦♥❡ ✐s t❤❛t ✇❡ ❝♦✉•❞ ♥♦t ♣r♦✈❡ t❤❛t ❛♥ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥ ❛•✇❛②s ❡①✐sts ✇❤❡♥ p < 1✳ ❚♦ s❦✐rt t❤✐s ♣r♦❜•❡♠✱ ✐♥ t❤❡ ♥❡①t ♣❛rt✱ ✇❡ ✇✐•• ♣r♦✈❡ t❤❛t t❤❡ ❛r❣✉♠❡♥t ❛•s♦ ✇♦r❦s ❢♦r ❞❡❝♦♠♣♦s✐t✐♦♥s t❤❛t ❛r❡ ❝•♦s❡ ❡♥♦✉❣❤ t♦ ❜❡✐♥❣ ♦♣t✐♠❛• ❜✉t ✐♥ t❤✐s ❝❛s❡ ✇❡ ♥❡❡❞ ❛♥ ❛❞❞✐t✐♦♥❛• ❝♦♥tr♦• ♦♥ t❤❡ ♦♣❡r❛t♦r ♥♦r♠ ♦❢ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✇❤✐❝❤ ✐s ❣✐✈❡♥ ❜② •❡♠♠❛ ✸✳✷✳✻✳ ❚❤❡ s❡❝♦♥❞ ❞✐✣❝✉•t② ✐s t❤❛t t♦ ✉s❡ ♣r♦♣♦s✐t♦♥ ✷✳✷✳✾✱ ✇❡ ♥❡❡❞ t♦ ✇♦r❦ ✐♥ ❛ ❞✐✛✉s❡ ❛•❣❡❜r❛✳ ❚♦ t❤❛t ❡✛❡❝t✱ ✇❡ s✐♠♣•② t❡♥s♦r ♦✉r ❜❛s❡ ❛•❣❡❜r❛ M ❜② L ∞ (0, 1) ✇❤✐❝❤ ✜①❡s t❤❡ ♣r♦♦❢ ✐♠♠❡❞✐❛t❡•② t❤❛♥❦s t♦ •❡♠♠❛ ✸✳✷✳✼✳ Pr♦♣♦s✐t✐♦♥ ✸✳✷✳✺✳ ❙✉♣♣♦s❡ t❤❛t M ✐s ❞✐✛✉s❡✳ ▲❡t p ∈ (0, 1]✱ ✐❢ x ∈ S(M) ❛♥❞ x = y + z ✐s ❛♥ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ L p t❤❡♥ ❢♦r ❛•• t ≥ 0✱ K t (Ry, p, ∞) ≤ A 2 p B p K t (Gx, p, ∞) ❛♥❞ K t (Cz, p, ∞) ≤ A 2 p B p K t (Gx, p, ∞)✳ Pr♦♦❢✳ ▲❡t e ❜❡ ❛ ♣r♦•❡❝t✐♦♥ ❝♦♠♠✉t✐♥❣ ✇✐t❤ |(Ry) * | ❛♥❞ f = 1 -e✳ ❚❛❦❡ ε > 0 ❛♥❞ y 1 , y 2 , z 1 , z 2 ∈ S(M) s✉❝❤ t❤❛t ex = y 1 + z 1 ✱ f x = y 2 + z 2 ✱ Ry 1 p p + ✻✻ ❈❍❆P❚❊| ✸✳ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ❑❍■◆❚❈❍■◆❊ ■◆❊◗❯❆▲■❚■❊❙ Cz 1 p p ≤ m p (ex) + ε ❛♥❞ Ry 2 p p + Cz 2 p p ≤ m p (f x) + ε✳ ❲❡ ❝❛♥ ✇r✐t❡ x = ex + f x = y 1 + z 1 + y 2 + z 2 ✳ ❚❤❡♥ ❜② ♠✐♥✐♠❛•✐t② ♦❢ y ❛♥❞ z✿ Ry p p + Cz p p ≤ R(y 1 + y 2 ) p p + C(z 1 + z 2 ) p p . ❲❡ ✉s❡ t❤❡ p✲tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t② ❛♥❞ ♦❜t❛✐♥ t❤❛t✿ R(y 1 + y 2 ) p p + C(z 1 + z 2 ) p p ≤ Ry 1 p p + Cz 1 p p + Ry 2 p p + Cz 2 p p . ❈♦♠❜✐♥❡❞ ✇✐t❤ •❡♠♠❛ 2.1.4✱ ✇❡ ❣❡t✿ R(ey) p p + R(f y) p p + Cz p p ≤ Ry 1 p p + Cz 1 p p + Ry 2 p p + Cz 2 p p . ◆♦✇ ✉s✐♥❣ t❤❡ ❛•♠♦st ♠✐♥✐♠❛•✐t② ♦❢ t❤❡ ❝♦✉♣•❡ (y 2 , z 2 ) ❛♥❞ •❡♠♠❛ 2.1.4✱ ✇❡ ♦❜t❛✐♥✿ Ry 2 p p + Cz 2 p p ≤ R(f y) p p + C(f z) p p + ε ≤ R(f y) p p + Cz p p + ε. ❍❡♥❝❡✱ ❝♦♠❜✐♥✐♥❣ t❤❡ •❛st t✇♦ ✐♥❡q✉❛•✐t✐❡s ❛♥❞ t❤❡♥ ✉s✐♥❣ t❤❡ ❑❤✐♥t❝❤✐♥❡ ✐♥✲ ❡q✉❛•✐t② ❢♦r p✿ R(ey) p p ≤ Ry 1 p p + Cz 1 p p + ε ≤ B p p G(ex) p p + 2ε. ❚❤✐s ✐s tr✉❡ ❢♦r ❛•• ε > 0 s♦✿ R(ey) p p ≤ B p p G(ex) p p . ❚❤❡ ♣r❡✈✐♦✉s ✐♥❡q✉❛•✐t② ❤♦•❞s ❢♦r ❛•• ♣r♦•❡❝t✐♦♥s e ❝♦♠♠✉t✐♥❣ ✇✐t❤ |(Ry) * |✳ ❚❛❦✐♥❣ t❤❡ s✉♣r❡♠✉♠ ♦✈❡r ❛•• s✉❝❤ e ✇✐t❤ τ (e) ≤ t p ❛♥❞ ✉s✐♥❣ ♣r♦♣♦s✐t✐♦♥ 2.2.9 ❛♥❞ r❡♠❛r❦ 2.2.10✱ ✇❡ ♦❜t❛✐♥✿ K t (Ry, p, ∞) ≤ A 2 p B p K t (Gx, p, ∞). ❚❤❡ ❝❛s❡ ♦❢ z ✐s ❡①❛❝t•② s②♠♠❡tr✐❝❛• ❜② t❛❦✐♥❣ ❛❞•♦✐♥ts ❛♥❞ s♦ t❤❡ ♣r♦♦❢ ✐s ❝♦♠♣•❡t❡✳ ❚♦ ♣r♦✈❡ t❤❡ t❤❡♦r❡♠ ✇✐t❤♦✉t ♠❛❦✐♥❣ ❛♥② ❛ss✉♠♣t✐♦♥s✱ t❤❡ ❢♦••♦✇✐♥❣ •❡♠♠❛ ✐s ❝r✉❝✐❛•✳ ▲❡♠♠❛ ✸✳✷✳✻✳ ▲❡t ε > 0 ❛♥❞ x ∈ S(M)✳ ❚❤❡r❡ ❡①✐st y ❛♥❞ z ✐♥ S(M) s✉❝❤ t❤❛t x = y + z✱ Ry p p + Cz p p ≤ m p (x) + ε✱ Ry ∞ ≤ 2 Gx ∞ ❛♥❞ Cz ∞ ≤ 2 Gx ∞ ✳ Pr♦♦❢✳ ❲✐t❤ t❤❡ ♥♦t❛t✐♦♥s ♦❢ t❤❡ •❡♠♠❛✱ ❝❤♦♦s❡ y, z ∈ S(M) s✉❝❤ t❤❛t y + z = x ❛♥❞ Ry p p + Cz p p ≤ m p (x) + ε ❛♥❞ ❞❡♥♦t❡ Gx ∞ = A✳ ▲❡t e = 1 [A,∞) (|(Ry) * |)✱ f = 1 [A,∞) (|Cz|)✳ ❲❡ ✇r✐t❡ x = e ⊥ xf ⊥ + ex + e ⊥ xf ❛♥❞ ❞❡❞✉❝❡ t❤❡ ♥❡✇ ❞❡❝♦♠♣♦s✐t✐♦♥✿ y ′ = e ⊥ yf ⊥ + ex ❛♥❞ z ′ = e ⊥ zf ⊥ + e ⊥ xf ✳ ▲❡t ✉s ❝❤❡❝❦ t❤❛t ✐t s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ •❡♠♠❛✳ ◆♦t❡ t❤❛t R(ex) ∞ ≤ R(x) ∞ ≤ A✳ ❚❤❡ •❛st ✐♥❡q✉❛•✐t② ❢♦••♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡ ξ i ❛r❡ ♦rt❤♦♥♦r♠❛• ❛♥❞ t❤❛t ❝♦♥s❡q✉❡♥t•② t❤❡ ♠❛♣ Id ⊗ τ A : M⊗A → M s❡♥❞s |(Gx) * | 2 t♦ |(Rx) * | 2 ✳ ▼♦r❡♦✈❡r✱ t❤❡ •❡❢t s✉♣♣♦rt ♦❢ R(ex) ✐s •❡ss t❤❛♥ e✱ ✐♥❞❡❡❞✿ |(R(ex)) * | 2 = e( i x i x * i )e✳ ❍❡♥❝❡ |R(ex) * | ≤ Ae✳ ◆♦t❡ ❛•s♦ t❤❛t |(R(ey)) * | ≥ Ae✳ ■♥❞❡❡❞✱ s✐♥❝❡ e ❛♥❞ |(Ry) * | ❝♦♠♠✉t❡ ❜② •❡♠♠❛ 2.1.4✱ |(R(ey)) * | = e |(Ry) * | ≥ Ae ❜② ❞❡✜♥✐t✐♦♥ ♦❢ e✳ ❇② s②♠♠❡tr②✱ ✇❡ ❤❛✈❡ t❤❡ s❛♠❡ ❦✐♥❞ ♦❢ ❡st✐♠❛t❡s ❢♦r t❤❡ ❝♦•✉♠♥s ✐✳❡ C(e ⊥ xf ) ≤ Af ❛♥❞ |C(zf )| ≥ Af ✳ ❋♦r r♦✇s✱ ✇❡ ❣❡t✿ Ry ′ p p ≤ R(e ⊥ yf ⊥ ) p p + R(ex) p p ≤ R(e ⊥ y) p p + τ (e)A p ≤ R(e ⊥ y) p p + R(ey) p p = Ry p p ✇❤❡r❡ t❤❡ •❛st ✐♥❡q✉❛•✐t② ✐s ❣✐✈❡♥ ❜② •❡♠♠❛ 2.1.4✳ ❙✐♠✐•❛r•②✱ ❢♦r ❝♦•✉♠♥s✱ ✇❡ ❣❡t✿ Cz ′ p p ≤ Cz p p . ❈♦♥s❡q✉❡♥t•②✿ Ry ′ p p + Cz ′ p p ≤ Ry p p + Cz p p ≤ m p (x) + ε. ❚❤❡ ❝♦♥tr♦• ✐♥ L ∞ ❛•s♦ ❢♦••♦✇s q✉✐❝❦•②✿ Ry ′ ∞ ≤ R(e ⊥ yf ⊥ ) ∞ + R(ex) ∞ ≤ R(e ⊥ y) ∞ + Rx ∞ ≤ 2A ✇❤❡r❡ t❤❡ •❛st ✐♥❡q✉❛•✐t② ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ e✳ ❚❤❡ ❝❛s❡ ♦❢ ❝♦•✉♠♥s ✐s ❛s ❛•✇❛②s s②♠♠❡tr✐❝❛• ✇❤✐❝❤ ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢✳ ❚❤❡ ❢♦••♦✇✐♥❣ •❡♠♠❛ ✐s t❤❡ ❦❡② t♦ r❡♠♦✈❡ t❤❡ ❤②♣♦t❤❡s✐s t❤❛t M ✐s ❞✐✛✉s❡✳ ▲❡♠♠❛ ✸✳✷✳✼✳ ❈♦♥s✐❞❡r t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ N = M⊗L ∞ ([0, 1]) ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ t❡♥s♦r ♣r♦❞✉❝t tr❛❝❡ ❛♥❞ ✐❞❡♥t✐❢② M ✇✐t❤ M ⊗ 1 ⊂ N ✳ ❚❤❡♥ ❢♦r ❛♥② p > 0 ❛♥❞ x ∈ S(M)✿ m p (x) = inf{ Rf p p + Cg p p : f + g = x, f, g ∈ S(N )}. Pr♦♦❢✳ ❙✐♥❝❡ M ⊂ N t❤❡ ✐♥❡q✉❛•✐t②✿ m p (x) ≥ inf{ Rf p p + Cg p p : f + g = x, f, g ∈ S(N )} ✐s ❝•❡❛r✳ ▲❡t f, g ∈ S(N ) s✉❝❤ t❤❛t f + g = x✳ ❙❡❡✐♥❣ f ❛♥❞ g ❛s ❢✉♥❝t✐♦♥s ❢r♦♠ [0, 1] t♦ S(M) ✇❡ ✇r✐t❡✿ Rf p p + Cg p p = 1 0 R(f (t)) p p + C(g(t)) p p dt ≥ 1 0 m p (x)dt = m p (x). ✻✽ ❈❍❆P❚❊| ✸✳ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ❑❍■◆❚❈❍■◆❊ ■◆❊◗❯❆▲■❚■❊❙ ❊①t❡♥❞ t❤❡ ♥♦t❛t✐♦♥ m p (h) t♦ ❡•❡♠❡♥ts h ∈ S(N ) ❜②✿ m p (h) := inf{ Rf p p + Cg p p : f + g = h, f, g ∈ S(N )}. ❲❤❡♥ x ∈ S(M) ✐s ❝♦♥s✐❞❡r❡❞ ❛s ❛♥ ❡•❡♠❡♥t ♦❢ S(N )✱ t❤❡r❡ ❛r❡ ♥♦✇ t✇♦ ❞❡✜♥✐t✐♦♥s ♦❢ m p (x)✳ ❊✐t❤❡r t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥s ❝❛♥ ❜❡ t❛❦❡♥ ✐s S(M) ♦r ✐♥ S(N )✳ ◆♦t❡ t❤❛t ❜② t❤❡ •❡♠♠❛ ❛❜♦✈❡✱ t❤❡s❡ t✇♦ ❞❡✜♥✐t✐♦♥s ❝♦✐♥❝✐❞❡✳ ✸✳✷✳✸ Pr♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠ ✐♥ ❢✉•• ❣❡♥❡r❛•✐t② ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♣r❡s❡♥t ❛ ♣r♦♦❢ ♦❢ t❤❡ ♠❛✐♥ r❡s✉•t ✉s✐♥❣ ❞❡❝♦♠♣♦s✐t✐♦♥s t❤❛t ❛r❡ ❝•♦s❡ t♦ ❜❡ ♦♣t✐♠❛• r❛t❤❡r t❤❛♥ ♦♣t✐♠❛•✳ ❲❡ ❡ss❡♥t✐❛••② ❢♦••♦✇ t❤❡ ♣r♦♦❢ ♦❢ ♣r♦♣♦s✐t✐♦♥ 3.2.5 ❜✉t ✇❡ ♥❡❡❞ s♦♠❡ ❛❞❞✐t✐♦♥❛• ❝❛r❡ ❛♥❞ •❡♠♠❛ 3.2.6 t♦ ❣❡t t❤❡ ✜♥❛• ❡st✐♠❛t❡✳ ❆♥ ❛•t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤ ✐s t♦ ✇♦r❦ ✐♥ ❛♥ ✉•tr❛♣r♦❞✉❝t ✇❤❡r❡ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥s ❛•✇❛②s ❡①✐st✱ ✐t ✇✐•• ❞❡✈❡•♦♣❡❞ ✐♥ t❤❡ ♥❡①t s✉❜s❡❝t✐♦♥✳ ◆♦t❡ t❤❛t t❤❡ t✇♦ str❛t❡❣✐❡s ②✐❡•❞ t❤❡ s❛♠❡ ❝♦♥st❛♥ts✳ ▲❡♠♠❛ ✸✳✷✳✽✳ ▲❡t 0 < p ≤ 1✳ ❋♦r ❛•• x ∈ S(M)✱ η > 0 ❛♥❞ ❞❡❝♦♠♣♦s✐t✐♦♥s x = y + z s✉❝❤ t❤❛t Ry p p + Cz p p ≤ m p (x) + η p ✱ ✇❡ ❤❛✈❡✿ K t (Ry, p, ∞) ≤ A p A p p B p p K t (Gx, p, ∞) p + η p 1/p ❢♦r ❛•• t > 0✳ Pr♦♦❢✳ ▲❡t p ∈ (0, ∞) ❛♥❞ x ∈ S(M)✳ ❚❛❦❡ η > 0 ❛♥❞ y, z ∈ S(M) s✉❝❤ t❤❛t y + z = x ❛♥❞ Ry p p + Cz p p ≤ m p (x) + η p ✳ ❚♦ ❜❡ ❛❜•❡ t♦ ✉s❡ •❡♠♠❛ 2.2.9✱ ✇❡ ♥❡❡❞ t♦ ✇♦r❦ ✐♥ ❛ ❞✐✛✉s❡ ❛•❣❡❜r❛ s♦ ✇❡ ✇✐•• ♥♦✇ ❝♦♥s✐❞❡r x, y ❛♥❞ z ❛s ❡•❡♠❡♥ts ♦❢ S(N )✳ ❲❡ ❝❛♥ ♥♦✇ r❡♣❡❛t t❤❡ ❛r❣✉♠❡♥t ♦❢ t❤❡ ♣r♦♦❢ ♦❢ ♣r♦♣♦s✐t✐♦♥ 3.2.5✳ ❚❛❦❡ e ❛ ♣r♦•❡❝t✐♦♥ ❝♦♠♠✉t✐♥❣ ✇✐t❤ |(Ry) * | ✐♥ S(N ) ❛♥❞ f = 1 -e✳ ❚❛❦❡ ε > 0 ❛♥❞ ex = y 1 + z 1 ✱ f x = y 2 + z 2 s✉❝❤ t❤❛t Ry 1 p p + Cz 1 p p ≤ m p (ex) + ε ❛♥❞ Ry 2 p p + Cz 2 p p ≤ m p (f x) + ε✳ ❇② ❞❡✜♥✐t✐♦♥ ♦❢ y ❛♥❞ z✱ Ry p p + Cz p p ≤ m p (x) + η p ✱ ❤❡♥❝❡ ✿ Ry p p + Cz p p ≤ R(y 1 + y 2 ) p p + C(z 1 + z 2 ) p p + η p . ❇② t❤❡ p✲tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t② ❛♥❞ •❡♠♠❛ 2.1.4✿ R(ey) p p + R(f y) p p + Cz p p ≤ Ry 1 p p + Cz 1 p p + Ry 2 p p + Cz 2 p p + η p . ◆♦✇ ✉s✐♥❣ t❤❡ ❛•♠♦st ♠✐♥✐♠❛•✐t② ♦❢ t❤❡ ❝♦✉♣•❡ (y 2 , z 2 ) ❛♥❞ •❡♠♠❛ 2.1.4✱ ✇❡ ♦❜t❛✐♥✿ Ry 2 p p + Cz 2 p p ≤ R(f y) p p + C(f z) p p + ε ≤ R(f y) p p + Cz p p + ε. ❆♥❞ ✇❡ ❝♦♥❝•✉❞❡ ✉s✐♥❣ t❤❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t②✿ R(ey) p p ≤ Ry 1 p p + Cz 1 p p + ε + η p ≤ B p p G(ex) p p + 2ε + η p . ✸✳✷✳ ❙❖▼❊ P|❖P❊|❚■❊❙ ❖❋ ❖P❚■▼❆▲ ❉❊❈❖▼P❖❙■❚■❖◆❙ ■◆ L P ❋❖| P < 2✻✾ ❚❤✐s ✐s tr✉❡ ❢♦r ❛•• ε > 0 s♦✿ R(ey) p p ≤ B p p G(ex) p p + η p . ❚❛❦✐♥❣ t❤❡ s✉♣r❡♠✉♠ ♦✈❡r ❛•• ♣r♦•❡❝t✐♦♥s e ∈ N ❝♦♠♠✉t✐♥❣ ✇✐t❤ |(Ry) * |✱ ❜② ♣r♦♣♦s✐t✐♦♥ 2.2.9✱ ✇❡ ♦❜t❛✐♥ t❤❛t✿ K t (Ry, p, ∞) ≤ A p (A p p B p p K t (Gx, p, ∞) p + η p ) 1/p . Pr♦♦❢ ♦❢ t❤❡♦r❡♠ 3.2.1✳ ▲❡t x ∈ S(M)✳ ❲❡ ♥❡❡❞ t♦ ✜♥❞ y ❛♥❞ z ✐♥ S(M) s✉❝❤ t❤❛t y + z = x✱ K t (Ry, p, ∞) K t (Gx, p, ∞) ❛♥❞ s✐♠✐•❛r•② ❢♦r Cz✳ ❲r✐t❡ ❛❣❛✐♥ A = Gx ∞ ✳ ❉❡✜♥❡ δ := τ (1 |Gx|>A/2 ) 1/p ✳ ❋♦r t < δ✱ ✭1.1✮ ❣✐✈❡s✿ A p K t (Gx, p, ∞) ≥ ( t p 0 µ u (Gx) p du) 1/p ≥ tA 2 . ▲❡t η = K δ (Gx, p, ∞)✳ ❚❛❦❡ y, z s✉❝❤ t❤❛t Ry p p + Cz p p ≤ m p (x) + η p ✱ Ry ∞ ≤ 2A ❛♥❞ Cz ∞ ≤ 2A✳ ❚❤✐s ✐s ♣♦ss✐❜•❡ ❜② •❡♠♠❛ 3.2.6✳ ❋♦r t ≥ δ✱ ✉s✐♥❣ •❡♠♠❛ 3.2.8 ❛♥❞ s✐♥❝❡ t → K t (Gx, p, ∞) ✐s ✐♥❝r❡❛s✐♥❣✱ ✇❡ ❤❛✈❡✿ K t (Ry, p, ∞) ≤ A p (A p p B p p K t (Gx, p, ∞) p + η p ) 1/p ≤ A p (A p p B p p K t (Gx, p, ∞) p + K δ (Gx, p, ∞) p ) 1/p ≤ A p (A p p B p p + 1) 1/p K t (Gx, p, ∞). ❋♦r t < δ✱ ✇❡ ❦♥♦✇ t❤❛t A p K t (Gx, p, ∞) ≥ tA 2 ❛♥❞ s✐♥❝❡ Ry ∞ ≤ 2A✱ ✇❡ ❛•s♦ ❤❛✈❡ K t (Ry, p, ∞) ≤ 2At s♦ K t (Ry, p, ∞) ≤ 4A p K t (Gx, p, ∞)✳ ▲❡t C p = max(A p (A p p B p p +1) 1/p , 4A p )✳ ❲❡ ❤❛✈❡ •✉st ♣r♦✈❡♥ t❤❛t ❢♦r ❛•• t > 0✱ K t (Ry, p, ∞) ≤ C p K t (Gx, p, ∞)✳ ❙✐♥❝❡ t❤❡ ❛r❣✉♠❡♥t ✐s ♣❡r❢❡❝t•② s②♠♠❡tr✐❝❛•✱ ✇❡ ❛•s♦ ❤❛✈❡ K t (Cz, p, ∞) ≤ C p K t (Gx, p, ∞)✳ |❡♠❛r❦ ✸✳✷✳✾✳ ❚❤❡ t❤❡♦r❡♠ st✐•• ❤♦•❞s ✇❤❡♥ p ∈ (1, 2]✳ ■♥❞❡❡❞✱ t❤❡ p✲ tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t② ✐s ❢❛•s❡ ✐♥ ❣❡♥❡r❛• ❜✉t ❤❡r❡✱ ✇❡ ♦♥•② ✉s❡ ✐t t♦ ♣r♦✈❡ ✐♥✲ ❡q✉❛•✐t✐❡s ♦❢ t❤❡ t②♣❡✿ ex + f y p p ≤ ex p p + f y p p ✭✸✳✷✮ ✇❤❡r❡ e ✐s ❛ ♣r♦•❡❝t✐♦♥✱ f = 1 -e ❛♥❞ x, y ∈ M✳ ❚❤✐s st✐•• ❤♦•❞s ❢♦r p ∈ [1, 2] ✉s✐♥❣ t❤❡ ✐♥❡q✉❛•✐t②✱ ❢♦r ❛•• a, b ∈ M✿ a -b p p + a + b p p 2 ≤ a p p + b p p . ✭✸✳✸✮ ✼✵ ❈❍❆P❚❊| ✸✳ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ❑❍■◆❚❈❍■◆❊ ■◆❊◗❯❆▲■❚■❊❙ ◆♦t❡ t❤❛t ex + f y = (e -f )(ex -f y) ❛♥❞ t❤❛t e -f ✐s ❛ ✉♥✐t❛r②✳ ❍❡♥❝❡ ex + f y p = ex -f y p ✳ ◆♦✇ t❛❦❡ a = ex ❛♥❞ b = f y ✐♥ (3.3) t♦ ♦❜t❛✐♥ (3.2)✳ ❚♦ ♣r♦✈❡ (3.3) ♦♥❡ ❝❛♥ ❢♦r ❡①❛♠♣•❡ ❛♣♣•② t❤❡ |✐❡s③✲❚❤♦r✐♥ ✐♥t❡r♣♦•❛t✐♦♥ t❤❡♦r❡♠ t♦ t❤❡ ❛♣♣•✐❝❛t✐♦♥ T : (x, y) → (x + y, x -y)✳ ✸✳✷✳✹ Pr♦♦❢ ✉s✐♥❣ ❛♥ ✉•tr❛♣r♦❞✉❝t ❛r❣✉♠❡♥t ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ♣r♦✈❡ t❤❡ ♠❛✐♥ r❡s✉•t ❜② r❡❞✉❝✐♥❣ ✐t t♦ ♣r♦♣♦s✐t✐♦♥ 3.2.5✳ ❲❡ ❤❛✈❡ ❛•r❡❛❞② s❡❡♥ t❤❛t ✇❡ ❝❛♥ ❣❡t r✐❞ ♦❢ t❤❡ ❤②♣♦t❤❡s✐s t❤❛t M ✐s ❞✐✛✉s❡ ❜② ❝♦♥s✐❞❡r✐♥❣ N = M⊗L ∞ ([0, 1])✳ ❍❡r❡✱ ✇❡ r❡❞✉❝❡ t❤❡ ♣r♦❜•❡♠ t♦ t❤❡ ❝❛s❡ ✇❤❡r❡ x ❤❛s ❛♥ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥ ❜② ♣•❛❝✐♥❣ ♦✉rs❡•✈❡s ✐♥ ❛♥ ✉•tr❛♣r♦❞✉❝t✳ ❲❡ ✇✐•• ❞❡♥♦t❡ ❜② M ω t❤❡ ✉•tr❛♣r♦❞✉❝t ♦❢ N ❝♦♣✐❡s ♦❢ M✳ ❙✐♥❝❡ ✇❡ ♦♥•② ❝♦♥s✐❞❡r s❡q✉❡♥❝❡s x ∈ S(M)✱ ❛s ♠❡♥t✐♦♥❡❞ ❜❡❢♦r❡✱ ♦♥❝❡ x ✐s ✜①❡❞✱ ✇❡ ❝❛♥ ✇♦r❦ ✐♥ (eMe) N ✇❤❡r❡ e = s(|(Rx) * |) ∨ s(|Cx|)✳ ❍❡♥❝❡✱ t♦ s✐♠♣•✐❢② t❤❡ ❡①♣♦s✐t✐♦♥ ♦❢ t❤✐s s✉❜s❡❝t✐♦♥✱ ✇❡ ✇✐•• s✉♣♣♦s❡ r✐❣❤t ❛✇❛② t❤❛t M ✐s ✜♥✐t❡✳ ❲❡ ❛•s♦ ✜① t❤❡ ❢♦••♦✇✐♥❣ ❝♦♥✈❡♥t✐♦♥✿ • ✐❢ x ∈ S(M)✱ s✐♥❝❡ M ✐s ❝❛♥♦♥✐❝❛••② ❡♠❜❡❞❞❡❞ ✐♥ M ω ✱ x ❝❛♥ ❛•s♦ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛♥ ❡•❡♠❡♥t ♦❢ S(M ω )✱ • ❝♦♥s✐❞❡r y ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ❡•❡♠❡♥ts ✐♥ B✱ y = (y (1) , ..., y (N ) ) ✇✐t❤ N ∈ N✳ ❋♦r i ≤ N ✱ ✇r✐t❡ y (i) = (y (i) n ) n∈N ❛♥❞ ❞❡✜♥❡✱ ❢♦r ❛•• n ∈ N✱ y n = (y (1)
n , ..., y

(N ) n ) ∈ S(M)✳ ▲❡♠♠❛ ✸✳✷✳✶✵✳ ▲❡t p > 0 ❛♥❞ x ∈ S(M)✳ ❚❤❡♥✿ m p (x) = inf{ Ry p p + Cz p p : x = y + z, y, z ∈ S(M ω )}.
❋✉rt❤❡r♠♦r❡✱ t❤❡r❡ ❡①✐sts y, z ✐♥ S(M ω ) s✉❝❤ t❤❛t m p (x) = Ry

p p + Cz p p ❛♥❞ r❡♣r❡s❡♥t❛t✐✈❡s y, z ∈ S(B) s✉❝❤ t❤❛t ❢♦r ❛•• n ∈ N✱ y n + z n = x✱ Ry n ∞ ≤ 2 Gx ∞ ❛♥❞ Rz n ∞ ≤ 2 Gx ∞ Pr♦♦❢✳ ▲❡t y, z ∈ S(M ω ) ❛♥❞ ❝❤♦♦s❡ r❡♣r❡s❡♥t❛t✐✈❡s y ❛♥❞ z ✐♥ B s✉❝❤ t❤❛t y n + z n = x ❢♦r ❛•• n ∈ N✳ Ry p p + Cz p p = lim ω τ (|Ry n | p ) + lim ω τ (|Cz n | p ) = lim ω τ (|Ry n | p ) + τ (|Cz n | p ) ≥ m p (x). ❚❤❡ ♦t❤❡r ✐♥❡q✉❛•✐t② ✐s ✐♠♠❡❞✐❛t❡ s✐♥❝❡ L p (M) ✐s ✐s♦♠❡tr✐❝❛••② ✐❞❡♥t✐✜❡❞ t♦ ❛ s✉❜s♣❛❝❡ ♦❢ L p (M ω )✳ ❚♦ ♣r♦✈❡ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ •❡♠♠❛ t❛❦❡ (y n , z n ) n≥0 ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ♦❢ ❡•❡♠❡♥ts ✐♥ S(M) s✉❝❤ t❤❛t ❢♦r ❛•• n ∈ N ✱ y n + z n = x✱ Ry n ∞ ≤ 2 Gx ∞ ✱ Rz n ∞ ≤ 2 Gx ∞ ❛♥❞ lim n→∞ Ry n p p + Cz n p p = m p (x)✳ ❚❤✐s ✐s ♣♦ss✐❜•❡ ✸✳✷✳ ❙❖▼❊ P|❖P❊|❚■❊❙ ❖❋ ❖P❚■▼❆▲ ❉❊❈❖▼P❖❙■❚■❖◆❙ ■◆ L P ❋❖| P < 2✼✶ t❤❛♥❦s t♦ •❡♠♠❛ 3.2.6✳ ❉❡♥♦t❡ ❜② y ❛♥❞ z t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡•❡♠❡♥ts ✐♥ S(M ω )✳ ❲❡ ❤❛✈❡ y + z = x ❛♥❞✿ Ry p p + Cz p p = lim ω τ (|Ry n | p ) + lim ω τ (|Cz n | p ) = lim ω τ (|Ry n | p ) + τ (|Cz n | p ) = m p (x). |❡♠❛r❦ ✸✳✷✳✶✶✳ ◆♦t❡ t❤❛t ❢♦r ❛•• p > 0✱ x ∈ M✱ lim t→∞ K t (x, ∞, p) = x ∞ ✳ ❆s ✉s✉❛•✱ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t x > 0✳ ❙✐♥❝❡ K(x, ∞, p) ✐s ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❜♦✉♥❞❡❞ ❜② x ∞ ✱ ✐t ❤❛s ❛ ✜♥✐t❡ •✐♠✐t l •❡ss ♦r ❡q✉❛• t❤❛♥ x ∞ ✳ ❙✉♣♣♦s❡ ❜② ❝♦♥tr❛❞✐❝✲ t✐♦♥ t❤❛t l < x ∞ ✳ ❚❤❡♥ ❢♦r ❛•• t✱ t❤❡r❡ ❡①✐sts y ∈ M s✉❝❤ t❤❛t y ∞ ≤ l ❛♥❞ x -y p ≤ l/t✳ ▲❡t e = 1 (l,∞) (x)✳ x -y p ≥ e(x -y)e p ✳ ◆♦✇ exe > el ❜② ❞❡✜♥✐t✐♦♥ ♦❢ e ❛♥❞ eye < el s♦ e(x -y)e p > e(x -l)e p ✳ ❍❡♥❝❡ ❢♦r ❛•• t✱ l/t ≥ e(x -l)e p > 0 ✇❤✐❝❤ ②✐❡•❞s ❛ ❝♦♥tr❛❝t✐❝t✐♦♥ t❛❦✐♥❣ •✐♠✐ts ✇❤❡♥ t ❣♦❡s t♦ ✐♥✜♥✐t②✳ Pr♦♦❢ ♦❢ t❤❡♦r❡♠ 3.2.1✳ ❇② t❤❡ ♣r❡✈✐♦✉s •❡♠♠❛ ❛♥❞ •❡♠♠❛ 3.2.7✱ ✇❡ ❝❛♥ ❛♣♣•② ♣r♦♣♦s✐t✐♦♥ 3.2.5 t♦ x ✐♥ t❤❡ ❛•❣❡❜r❛ N = M ω ⊗L ∞ ([0, 1])✳ ❍❡♥❝❡✱ ❛♥ ♦♣t✐✲ ♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥ y, z ❣✐✈❡♥ ❜② t❤❡ ♣r❡✈✐♦✉s •❡♠♠❛ ✈❡r✐✜❡s✱ ❢♦r ❛•• t ≥ 0✱ K t (Ry, p, ∞) ≤ A 2 p B p K t (Gx, p, ∞) ❛♥❞ K t (Cz, p, ∞) ≤ A 2 p B p K t (Gx, p, ∞)✳ ◆♦✇✱ ❜② •❡♠♠❛ 2.3.1✱ ❛♣♣•✐❡❞ t♦ |Ry * |✱ ✇❡ ❦♥♦✇ t❤❛t K(Ry n , L p (M), M) ❝♦♥✲ ✈❡r❣❡s ✉♥✐❢♦r♠•② t♦ K(Ry, L p (M ω ), M ω )✳ ◆♦t❡ t❤❛t lim t→∞ K t (Gx, ∞, p) = Gx ∞ s♦ t❤❡r❡ ❡①✐sts t 0 ∈ R + s✉❝❤ t❤❛t ❢♦r ❛•• t ≥ t 0 ✱ K t (Gx, ∞, p) ≥ 1 2 Gx ∞ ✳ ◆♦✇ •❡t ε = K 1/t0 (Ry, L p (M ω ), M ω ) ❛♥❞ ❝❤♦s❡ N s✉❝❤ t❤❛t ❢♦r ❛•• t ≥ 0✱ |K t (Ry N , L p (M), M) -K t (Ry, L p (M ω ), M ω )| ≤ ε ❙✐♥❝❡ t❤❡ K✲❢✉♥❝t✐♦♥♥❛• ✐s ✐♥❝r❡❛s✐♥❣ t❤✐s ♠❡❛♥s t❤❛t ❢♦r ❛•• t ≥ 1/t 0 ✱ K t (Ry N , L p (M), ∞) ≤ 2K t (Ry, L p (M ω ), M ω )✳ ◆♦✇✱ ♥♦t❡ t❤❛t K t (y N , L p (M), M) = tK 1/t (Ry N , M, L p (M)) ❛♥❞ r❡❝❛•• t❤❛t Ry N ∞ ≤ 2 Gx ∞ ✳ ❋♦r t < 1/t 0 ✿ K t (y N , L p (M), M) = tK 1/t (Ry N , M, L p (M)) ≤ t Ry N ∞ ≤ 2t Gx ∞ ≤ 4tK 1/t (Gx, ∞, p) ≤ 4K t (Gx, p, ∞). ◆♦✇✱ ❜② ❛♣♣•②✐♥❣ t❤❡♦r❡♠ 2.2.6✱ ✇❡ ❦♥♦✇ t❤❛t K t (Ry N , L p (M), M) ≈ K t (Ry N , p, ∞) ❛♥❞ t❤❛t K t (Ry N , L p (M ω ), M ω ) ≈ K t (Ry, p, ∞)✳ ❙♦✱ ❢♦r t ≤ 1/t 0 ✿ K t (Ry N , p, ∞) ≤ c p K t (Ry N , L p (M), M) ≤ 4c p K t (Gx, p, ∞). ✼✷ ❈❍❆P❚❊| ✸✳ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ❑❍■◆❚❈❍■◆❊ ■◆❊◗❯❆▲■❚■❊❙ ❆♥❞ ❢♦r t ≥ 1/t 0 ✿ K t (Ry N , p, ∞) ≤ c p K t (Ry N , L p (M), M) ≤ 2c p K t (Ry, L p (M ω ), M ω ) ≤ 2c 2 p K t (Ry, p, ∞) ≤ 2c 2 p A 2 p B p K t (Gx, p, ∞). ✸✳✸ ❋✉rt❤❡r r❡s✉•ts ♦♥ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥s ✐♥ L 1 ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✐♥✈❡st✐❣❛t❡ ❢✉rt❤❡r t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥s ✐♥ L 1 ✳ ❚❤❡ ✜rst ♥♦t❛❜•❡ ❢❛❝t ✐s t❤❛t ✐♥ t❤✐s ❝❛s❡✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t ❛♥ ♦♣✲ t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥ ❛•✇❛②s ❡①✐sts ✭s❡❡ •❡♠♠❛ ✸✳✸✳✶✮✳ ❑♥♦✇✐♥❣ t❤✐s✱ ❛ s✐♠♣•❡ ❞✉❛•✐t② ❛r❣✉♠❡♥t ②✐❡•❞s ❛ ❢❛❝t♦r✐s❛t✐♦♥ ❢♦r ❡•❡♠❡♥ts ✐♥ S(M) ✭t❤❡♦r❡♠ ✸✳✸✳✷✮✳ |❡♠❛r❦❛❜•②✱ t❤✐s r❡s✉•t ✐s ♦❢ ♣✉r❡•② ❛•❣❡❜r❛✐❝ ♥❛t✉r❡ ❛♥❞ ❝♦♠❜✐♥❡❞ ✇✐t❤ ❬✺✹( ✇❤✐❝❤ ♣r♦✈✐❞❡s t❤❡ ♥❡❝❡ss❛r② ❡st✐♠❛t❡ ♦♥ ❛♥t✐❝♦♠♠✉t❛t♦rs✱ ♣r♦❞✉❝❡s ❛ ♥❡✇ ♣r♦♦❢ ♦❢ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s✳ ❚❤❡ ♠❛✐♥ ♥♦✈❡•t② ✐s t❤❡ ❡♠❡r❣❡♥❝❡ ♦❢ ❡•❡♠❡♥ts α, β ∈ M + ❛ss♦❝✐❛t❡❞ t♦ x ∈ S(M) ✇❤✐❝❤ ♣•❛② t❤❡ r♦•❡ ♦❢ ❛ ✧♠♦❞✉•✉s✧ ✐♥ H p ✲s♣❛❝❡s ✭t❤❡♦r❡♠ ✸✳✶✳✽✮✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡r❡ ✐s ♥♦ ♥❡❡❞ ✐♥ t❤❡ ♣r♦♦❢s t♦ ❞✐s✲ t✐♥❣✉✐s❤ ❜❡t✇❡❡♥ p ≤ 2 ♦r p ≥ 2✳ ❚❤❡ ❞r❛✇❜❛❝❦ ♦❢ t❤❡ ♠❡t❤♦❞ ✐s t❤❛t ✐t r❡•✐❡s ♦♥ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ L ∞ ❛♥❞ t❤❡r❡❢♦r❡ ❞♦❡s ♥♦t ❛♣♣•②✱ ❛t •❡❛st ❞✐r❡❝t•②✱ t♦ |❛❞❡♠❛❝❤❡r ✈❛r✐❛❜•❡s✳ ▲❡t ✉s ❛•r❡❛❞② ❛ss✉♠❡ t❤❛t ❢♦r ❛•• x ∈ S(M)✱ Gx ∞ c∞ max( Rx ∞ , Cx ∞ ). ✭✸✳✹✮ ❲❡ st❛rt ❜② ♣r♦✈✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ♦♣t✐♠❛• ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ L 1 ✳ ❚❤❡ ❛r❣✉♠❡♥t ✐s str❛✐❣❤t❢♦r✇❛r❞ ❜② t❛❦✐♥❣ ❛ •✐♠✐t ♦❢ ❛ ♠✐♥✐♠✐s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❞❡✲ ❝♦♠♣♦s✐t✐♦♥s✳ ▲❡♠♠❛ ✸✳✸✳✶✳ ▲❡t x ∈ S(M)✳ ❚❤❡r❡ ❡①✐st y, z ∈ S(M) s✉❝❤ t❤❛t y + z = x ❛♥❞ Ry 1 + Cz 1 = m 1 (x)✳ Pr♦♦❢✳ ❈♦♥s✐❞❡r ❛ s❡q✉❡♥❝❡ (y (i) , z (i) ) i≥0 s✉❝❤ t❤❛t✿ lim i→∞ Ry (i) 1 + Cz (i) 1 = m 1 (x), ❛♥❞ ❢♦r ❛•• i ≥ 0✱ y (i) + z (i) = x✳ ❇② •❡♠♠❛ 3.2.6✱ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t t❤❡ y (i) n ❛♥❞ z (i) n ❛r❡ ✉♥✐❢♦r♠•② ❜♦✉♥❞❡❞ ✐♥ L ∞ (M)✳ |❡❝❛•• t❤❛t t❤❡ s❡q✉❡♥❝❡ x ✐s ✜♥✐t❡✱ s❛② ♦❢ •❡♥❣t❤ N ❛♥❞ •❡t✿ e = s(|(Rx) * |) ∨ s(|Cx|). ❇② ❝♦♥s✐❞❡r✐♥❣ t❤❡ s❡q✉❡♥❝❡s (ey (i) n e) n≤N ❛♥❞ (ez (i) e) n≤N ✇❡ ❝❛♥ ❛•s♦ s✉♣♣♦s❡ t❤❛t y (i) ❛♥❞ z (i) ❛r❡ ✐♥ (eMe) N ✳ ❙✐♥❝❡ t❤❡ y (i) n ❛♥❞ z (i) n ❛r❡ ✉♥✐❢♦r♠•② ❜♦✉♥❞❡❞ ✐♥ L ∞ (eMe)✱ ❜② t❤❡ ❝r✐t❡r✐♦♥ ♦❢ ✇❡❛❦ ❝♦♠♣❛❝✐t② ✐♥ L 1 ✭❬✺✽(✮✱ ✉♣ t♦ ❡①tr❛❝t✐♦♥✱ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t ❢♦r ❛•• n ≥ 0✱ t❤❡ s❡q✉❡♥❝❡ (y (i) n ) i≥0 ❝♦♥✈❡r❣❡s ✇❡❛❦•② ✐♥ L 1 (eMe) t♦ ❛♥ ❡•❡♠❡♥t y n ∈ eMe ❛♥❞ ✉s✐♥❣ ▼❛③✉r✬s •❡♠♠❛✱ t❛❦✐♥❣ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥s ✇❡ ❝❛♥ ❡✈❡♥ ❛ss✉♠❡ t❤❡ ♥♦r♠✲❝♦♥✈❡r❣❡♥❝❡ ✐♥ L 1 (M)✳ ▲❡t y = (y n ) 0≤n≤N ❛♥❞ z = (z n ) 0≤n≤N ✳ ❙✐♥❝❡ t❤❡ s❡q✉❡♥❝❡s y ❛♥❞ z ❜❡•♦♥❣ t♦ (eMe) N ✱ t❤❡② ❜❡•♦♥❣ t♦ S(M)✳ ◆♦t❡ t❤❛t {a, b : a + b = x} ✐s ❝•♦s❡❞ ❛♥❞ ❝♦♥✈❡①✱ s♦ ❢♦r ❛•• n ≥ 0✱ y n + z n = x n ✳ ▼♦r❡♦✈❡r✱ ❢♦r 0 ≤ n ≤ N ✱ ✇❡ ❤❛✈❡ lim i→∞ y (i) n ⊗ e 1,n = y n ⊗ e 1,n ✐♥ L 1 (eMe⊗B(ℓ 2 )) ❛♥❞ s✐♠✐•❛r•② lim i→∞ z (i) n ⊗ e n,1 = z n ⊗ e n,1 ✳ ❍❡♥❝❡✱ ❜② s✉♠♠✐♥❣ ♦✈❡r n✱ ✇❡ ♦❜t❛✐♥✿ m 1 (x) = lim i→∞ Ry (i) 1 + Cz (i) 1 = Ry 1 + Cz 1 ✳ ❚❤❡♦r❡♠ ✸✳✸✳✷✳ ▲❡t x ∈ S(M)✳ ❚❤❡r❡ ❡①✐st α, β ∈ M + c ❛♥❞ u ∈ S(M) s✉❝❤ t❤❛t x = αu + uβ✱ s(α) ≤ |(Ru) * | ≤ 1 ❛♥❞ s(β) ≤ |Cu| ≤ 1✳ Pr♦♦❢✳ ❯s✐♥❣ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥s ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣r♦♦❢✱ ✇❡ ❝❛♥ ✇♦r❦ ✐♥ (eMe) N ✇❤✐❝❤ ❣❛r❛♥t❡❡s t❤❛t ❛•• s❡q✉❡♥❝❡s ❝♦♥s✐❞❡r❡❞ ❛r❡ ✐♥ S(M) ❛♥❞ ♦♣✲ ❡r❛t♦rs ❛r❡ ✜♥✐t❡•② s✉♣♣♦rt❡❞✳ ▲❡t y, z ∈ S(M) ❜❡ t❤❡ ❡•❡♠❡♥ts ❣✐✈❡♥ ❜② •❡♠♠❛ ✸✳✸✳✶ ✐✳❡ y + z = x ❛♥❞ Ry 1 + Cz 1 = m 1 (x)✳ ❉❡♥♦t❡ α = |(Ry) * |✱ e = s(α)✱ β = |Cz| ❛♥❞ f = s(β)✳ ❲r✐t❡ y = αv✱ v ∈ S(M) s✉❝❤ t❤❛t αRv ✐s ❛ ♣♦•❛r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ Ry✱ ✐♥ ♣❛rt✐❝✉•❛r v ❝❛♥ ❜❡ ❝❤♦s❡♥ s✉❝❤ t❤❛t ev = v✳ ❙✐♠✐•❛r•②✱ z = wβ ✇✐t❤ w ∈ S(M) ❛♥❞ wf = w✳ ❇② ❞✉❛•✐t②✱ t❤❡r❡ ❡①✐sts ❛♥ ❡•❡♠❡♥t u ∈ (eMe) N s✉❝❤ t❤❛t u R∞∩C∞ = 1 ❛♥❞ τ ( i≥0 u * i x i ) = x R1+C1 = m 1 (x).
▲❡t ✉s r❡✇r✐t❡ t❤❡ ♣r❡✈✐♦✉s ❡q✉❛•✐t② ✇✐t❤ t❤❡ ♥♦t❛t✐♦♥s ✐♥tr♦❞✉❝❡❞ ♣r❡✈✐♦✉s•②✿

m 1 (x) = τ i≥0 u * i (αv i + w i β) τ (α) + τ (β) = τ i≥0 αv i u * i e + τ i≥0 f u * i w i β t❛❦✐♥❣ t❤❡ r❡❛• ♣❛rt ♦♥ ❜♦t❤ s✐❞❡s ♦❢ t❤❡ ❡q✉❛•✐t②✱ ✇❡ ❣❡t✿ τ (α) + τ (β) = τ ℜ( i≥0 αv i u * i e) + τ ℜ( i≥0 f u * i w i β) ❛♥❞ ❜② t❤❡ tr❛❝✐❛• ♣r♦♣❡rt② ♦❢ τ ✿ τ (α) + τ (β) = τ αℜ( i≥0 v i u * i e) + τ ℜ( i≥0 f u * i w i )β) . ▼♦r❡♦✈❡r✱ ℜ( i≥0 v i u * i e) ∞ = ℜ(R(v)C(u * e)) ∞ ≤ 1 ❛♥❞ s✐♥❝❡ ev = v✿ s(ℜ( i≥0 v i u * i e)) = s(ℜ(e( i≥0 v i u * i )e)) = s(eℜ( i≥0 v i u * i )e) ≤ e ✼✹ ❈❍❆P❚❊| ✸✳ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ❑❍■◆❚❈❍■◆❊ ■◆❊◗❯❆▲■❚■❊❙ s♦ ℜ( i≥0 v i u * i e) ≤ e✳ ❙✐♠✐•❛r•②✱ s✐♥❝❡ wf = w✱ ℜ( i≥0 f u * i w i ) ≤ f. ❍❡♥❝❡✱ ✇❡ ♠✉st ❤❛✈❡ ℜ( i≥0 v i u * i e) = e.
❚❤✐s ♠❡❛♥s t❤❛t

|τ (R(v)C(u * e))| = R(v) 2 C(u * e) 2 . ❚❤❡r❡ ✐s ❡q✉❛•✐t② ✐♥ t❤❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛•✐t② ✐♥ L 2 (M⊗B(ℓ 2 ))✱ s♦ t❤❡r❡ ❡①✐sts λ ∈ C✱ eu = λv✳ ❚❤❡ ♦♥•② ♣♦ss✐❜✐•✐t② ✐s t❤❛t eu = v✳ ❍❡♥❝❡✱ αu = αeu = αv = y✳ ❙✐♠✐•❛r•②✱ uβ = z✳ ❚❤❡r❡❢♦r❡✱ x = αu + uβ✳ ▲❡t ✉s ♥♦✇ ✈❡r✐❢② t❤❡ ♦t❤❡r r❡q✉✐r❡❞ ♣r♦♣❡rt✐❡s✳ ❋✐rst✱ s✐♥❝❡ u R∞∩C∞ = 1✱ |(Ru) * | ≤ 1✳ ❙❡❝♦♥❞•②✱ s✐♥❝❡ eu = v✱ e = |(R(eu)) * | ❛♥❞ ♥♦t❡ t❤❛t Ru ∞ ≤ 1 ✐♠♣•✐❡s t❤❛t e(Ru)(Ru) * 2 2 ≤ τ (e)✳ ▼♦r❡♦✈❡r✱ ❜② ♦rt❤♦❣♦♥❛•✐t②✱ e(Ru)(Ru) * 2 2 = e(Ru)(Ru) * e 2 2 + e(Ru)(Ru) * (1-e) 2 2 . ❍❡♥❝❡✱ τ (e) ≥ τ (e)+ e(Ru)(Ru) * (1- e) 2 2 ✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t e(Ru) * (Ru)(1-e) = 0✳ ❙②♠♠❡tr✐❝❛••②✱ (1-e)(Ru)(Ru) * e = 0 s♦ (Ru)(Ru) * -e = (Ru)(Ru) * -e(Ru)(Ru) * e = (1 -e)(Ru) * (Ru)(1 -e) ≥ 0. ❇② ❛❞•✉♥❝t✐♦♥✱ ✇❡ ♦❜t❛✐♥ s✐♠✐•❛r ✐♥❡q✉❛•✐t✐❡s ❢♦r ❝♦•✉♠♥s✳ ❲❡ ❛r❡ ♥♦✇ ❣♦✐♥❣ t♦ s❤♦✇ t❤❛t ✇❡ ❝❛♥ ♦❜t❛✐♥ ❑❤✐♥t❝❤✐♥❡✲t②♣❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ ❛ ✈❡r② ❣❡♥❡r❛• s❡♥s❡ ❢r♦♠ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ❢♦✉♥❞ ❛❜♦✈❡✳ ❚❤❡♦r❡♠ ✸✳✸✳✸✳ ▲❡t x ∈ S(M)✱ α, β ∈ M + c ❛♥❞ u ∈ S(M) s✉❝❤ t❤❛t x = αu + uβ✱ s(α) ≤ |(Ru) * | ≤ 1 ❛♥❞ s(β) ≤ |Cu| ≤ 1✳ ❚❤❡♥✱ ❢♦r ❛•• t > 0 ❛♥❞ p > 0✿ K t (α, p, ∞) + K t (β, p, ∞) ≈ c K t (Gx, p, ∞), ✇❤❡r❡ c ♦♥•② ❞❡♣❡♥❞s ♦♥ p ❛♥❞ (ξ i ) ✭♠♦r❡ ♣r❡❝✐s❡•② ♦♥ t❤❡ ❝♦♥st❛♥t c ∞ ❛♣♣❡❛r✐♥❣ ✐♥ ✭✸✳✹✮✮✳ Pr♦♦❢✳ ❯♣♣❡r ❡st✐♠❛t❡✳ ▲❡t p > 0 ❛♥❞ t > 0✳ ◆♦t❡ t❤❛t Gx = α(Gu) + (Gu)β ✇❤❡r❡ α ✐s ✐❞❡♥t✐✜❡❞ ✇✐t❤ α ⊗ 1 ❛♥❞ β ✇✐t❤ β ⊗ 1 ✐♥ M⊗A✳ ❇② t❤❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✐♥ L ∞ ✭✐✳❡✳ ❡st✐♠❛t❡ ✭✸✳✹✮✮✱ s✐♥❝❡ u R∞∩C∞ ≤ 1 ✇❡ ❤❛✈❡ Gu ∞ ≤ c ∞ ✳ ❍❡♥❝❡✱ K t (Gx, p, ∞) K t (α(Gu), p, ∞) + K t ((Gu)β, p, ∞) ≤ c ∞ (K t (α, p, ∞) + K t (β, p, ∞)). ▲♦✇❡r ❡st✐♠❛t❡✳ ▲❡t t > 0✳ ❲❡ ♦♥•② ♣r♦✈❡ t❤❡ t❤❡♦r❡♠ ❢♦r p < 1✱ t♦ ♦❜t❛✐♥ t❤❡ r❡s✉•t ❢♦r p ≥ 1 ✐t s✉✣❝❡s t♦ t❛❦❡ θ = 1 ✐♥ t❤❡ ❛r❣✉♠❡♥t✳ ▲❡t ✉s r❡✇r✐t❡ •❡♠♠❛ ✷✳✹✳✶✱ ✇✐t❤ θ = p✱ q = ∞✿ K t p (α p (Gu) + (Gu)β p , 1, ∞) K t (Gx, p, ∞) p Gu 1-p ∞ . ✸✳✸✳ ❋❯|❚❍❊| |❊❙❯▲❚❙ ❖◆ ❖P❚■▼❆▲ ❉❊❈❖▼P❖❙■❚■❖◆❙ ■◆ L 1 ✼✺ ❲✐t❤ t❤✐s ✐♥❡q✉❛•✐t②✱ ✇❡ ❝❛♥ ❝♦♥❝•✉❞❡ ✇✐t❤♦✉t t♦♦ ♠✉❝❤ ❡✛♦rt✳ ■♥❞❡❡❞✿ K t p (α p (Gu) + (Gu)β p , 1, ∞) ≥ Gu -1 ∞ K t p (((Gu)β p + α p (Gu))(Gu) * , 1, ∞). ❇② ✭✸✳✹✮✱ Gu ∞ u R∞∩C∞ = 1✳ ▲❡t ✉s ♥♦✇ ❝♦♥s✐❞❡r t❤❡ ❝♦♥❞✐t✐♦♥❛• ❡①♣❡❝t❛t✐♦♥ Id ⊗ τ A ✳ (Id ⊗ τ A )(((Gu)β p + α p (Gu))(Gu) * ) = (Id ⊗ τ A )( i,j≥0 u i β p u * j ⊗ ξ i ξ * j + i,j≥0 α p u i u * j ⊗ ξ i ξ * j ) = ∞ i=0 u i β p u * i + ∞ i=0 α p u i u * i . ❆s ❛ ❝♦♥❞✐t✐♦♥❛• ❡①♣❡❝t❛t✐♦♥✱ Id ⊗ τ A ✐s ❜♦✉♥❞❡❞ ♦♥ L 1 ❛♥❞ L ∞ s♦ K t p (((Gu)β p + α p (Gu))(Gu) * , 1, ∞) K t p ( ∞ i=0 u i β p u * i + α p ( ∞ i=0 u i u * i ), 1, ∞), ♥♦t❡ t❤❛t ∞ i=0 u i β p u * i ≥ 0✱ ❛♥❞ t❤❛t s✐♥❝❡ s(α) ≤ ∞ i=0 u i u * i ≤ 1✱ ✇❡ ❣❡t α p ( ∞ i=0 u i u * i ) = α p ≥ 0✳ ❍❡♥❝❡✱ K t p ( ∞ i=0 u i β p u * i + α p ( ∞ i=0 u i u * i ), 1, ∞) K t p (α p , 1, ∞) K t (α, p, ∞) p , ✇❤❡r❡ ✇❡ ✉s❡❞ t❤❡ ♣♦✇❡r t❤❡♦r❡♠ ✭♣r♦♣♦s✐t✐♦♥ ✷✳✷✳✶✮ ❢♦r t❤❡ •❛st ✐♥❡q✉❛•✐t②✳ ❚❤❡ s❛♠❡ tr✐❝❦s ✇♦r❦ ❢♦r β ❜② ♠✉•t✐♣•②✐♥❣ Gx ❜② (Gu) * ♦♥ t❤❡ •❡❢t✳ |❡♠❛r❦ ✸✳✸✳✹✳ ❚❤❡♦r❡♠ ✸✳✶✳✽ ❝•❛✐♠❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ✐s ♦❜t❛✐♥❡❞ ❜② ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ t✇♦ ♣r❡✈✐♦✉s t❤❡♦r❡♠s ❛♥❞ t❤❡ ❝❤❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ ✐♥t❡r♣♦✲ •❛t✐♦♥ s♣❛❝❡s ❜❡t✇❡❡♥ L p ✲s♣❛❝❡s ❣✐✈❡♥ ❜② ♣r♦♣♦s✐t✐♦♥ ✶✳✸✳✺✳ |❡♠❛r❦ ✸✳✸✳✺✳ ■❢ ✇❡ st❛rt ✇✐t❤ ❛♥ x ∈ S(M) s✉❝❤ t❤❛t ❢♦r ❛•• i ≥ 0✱ x i = x * i ✱ t❤❡ ❢❛❝t♦r✐s❛t✐♦♥ ❣✐✈❡♥ ❜② t❤❡♦r❡♠ ✸✳✸✳✷ t❛❦❡s t❤❡ ❢♦••♦✇✐♥❣ ❢♦r♠✳ ❚❤❡r❡ ❡①✐sts α ∈ M + ❛♥❞ u ∈ S(M) s✉❝❤ t❤❛t • x = uα + αu✱ • ❢♦r ❛•• i ≥ 0✱ u i = u * i ✱ • s(α) ≤ i≥0 u 2 i ≤ 1✳ |❡♠❛r❦ ✸✳✸✳✻✳ ❋r♦♠ t❤❡ r❡s✉•ts ♦❢ t❤✐s s❡❝t✐♦♥✱ ✐t ✐s ❡❛s② t♦ r❡❝♦✈❡r t❤❡ ✉♣♣❡r ❛♥❞ •♦✇❡r ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✐❢ E ✐s ❛♥ ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ ❜❡t✇❡❡♥ L p ✲s♣❛❝❡s✱ p ∈ (0, ∞] ❛♥❞ x ∈ S(M)✿ x R E +C E Gx E(M A ) x R E ∩C E . Pr♦♦❢✳ ▲❡t x ∈ S(M)✳ ❚❤❡ •❡❢t ❤❛♥❞ s✐❞❡ ✐♥❡q✉❛•✐t② ✐s ♦❜t❛✐♥❡❞ ❞✐r❡❝t•② ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ y = αu ❛♥❞ z = uβ ❛♥❞ ❛♣♣•②✐♥❣ t❤❡ •♦✇❡r ❡st✐♠❛t❡ ✐♥ t❤❡♦r❡♠ ✸✳✶✳✽✳ ❚♦ s❤♦✇ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ✐♥❡q✉❛•✐t②✱ ✇❡ ♠❛❦❡ ❛ ❝♦♠♣✉t❛t✐♦♥ s✐♠✐•❛r t♦ ✇❤❛t ❛♣♣❡❛r❡❞ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡♦r❡♠ ✸✳✸✳✸✳ ❋✐rst✱ ✉s✐♥❣ ❛❣❛✐♥ t❤❡♦r❡♠ ✸✳✶✳✽✱ ✇❡ ❦♥♦✇ t❤❛t Gx E(M A ) max( α E(M) , β E(M) ). ✼✻ ❈❍❆P❚❊| ✸✳ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ❑❍■◆❚❈❍■◆❊ ■◆❊◗❯❆▲■❚■❊❙ ▼♦r❡♦✈❡r✱ ♥♦t❡ t❤❛t Rx E(M B ) Rx(Ru) * E(M B ) = α + (Ru)β(Ru) * E(M B ) ≥ α E(M) . ❇② ❛❞•✉♥❝t✐♦♥✱ ✇❡ ❛•s♦ ♦❜t❛✐♥ Cx E(M B ) β E(M) . ❍❡♥❝❡✱ Gx E(M A ) max Rx E(M B ) , Cx E(M B ) . ✸✳✹ ❆ r❡♠❛r❦ ❛❜♦✉t ♠❛rt✐♥❣❛•❡ ✐♥❡q✉❛•✐t✐❡s ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ r❡❝♦✈❡r ❛ ✈❛r✐❛♥t ♦❢ ❛ r❡s✉•t ✜rst ♣r♦✈❡❞ ✐♥ ❬✺✷( ♦♥ ♠❛rt✐♥❣❛•❡ ✐♥❡q✉❛•✐t✐❡s✳ ❚❤❡ ♥♦✈❡•t② ❝♦♠♣❛r❡❞ t♦ t❤❡ ♦r✐❣✐♥❛• ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❛rt✐♥❣❛•❡ ✐♥❡q✉❛•✐t✐❡s ✭❬✹✾(✮ ✐s t❤❛t ✇❡ s❤♦✇ t❤❛t t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♣♣❡❛r✐♥❣ ❢♦r 1 < p < 2 ❝❛♥ ❜❡ ❝❤♦s❡♥ t♦ ❜❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ p✳ ◆♦t❡ t❤❛t t❤✐s r❡s✉•t ❤❛s ❛•s♦ ❜❡❡♥ ♦❜t❛✐♥❡❞✱ ✉s✐♥❣ ❛ ❝♦♥str✉❝t✐✈❡ ❛♣♣r♦❛❝❤✱ ✐♥ ❛ r❡❝❡♥t ♣r❡♣r✐♥t ✭❬✷✸(✮✳ ❚❤❡ s❡tt✐♥❣ ✐s t❤❡ ❢♦••♦✇✐♥❣✱ •❡t F = (M n ) n≥0 ❜❡ ❛ ✜•tr❛t✐♦♥ ♦♥ M✱ ❛♥❞ ❛ss✉♠❡ t❤❛t ❢♦r ❛•• n ∈ N✱ t❤❡ ❝♦♥❞✐t✐♦♥❛• ❡①♣❡❝t❛t✐♦♥ E n : M → M n ❡①✐sts✳ ❉❡♥♦t❡ ❜② M - ∞ := ∪ n≥0 M n t❤❡ s❡t ♦❢ ✜♥✐t❡ ❜♦✉♥❞❡❞ ♠❛rt✐♥❣❛•❡s ❢♦r t❤❡ ✜•tr❛t✐♦♥ F✳ ❋♦r ❛♥② x ✐♥ M - ∞ ✱ ❞❡♥♦t❡ ❜② dx t❤❡ ❛ss♦❝✐❛t❡❞ ♠❛rt✐♥❣❛•❡ ❞✐✛❡r❡♥❝❡s✳ ❋♦r t❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤❡ s❡❝t✐♦♥✱ s✉♣♣♦s❡ t❤❛t t❤❡ ξ i ❛r❡ ✐♥❞❡♣❡♥❞❡♥t |❛❞❡♠❛❝❤❡r ✈❛r✐❛❜•❡s✳ ❚❤❡♦r❡♠ ✸✳✹✳✶✳ ❚❤❡r❡ ❡①✐st ❝♦♥st❛♥ts (k p ) p>1 s✉❝❤ t❤❛t ❢♦r ❛♥② x ∈ M - ∞ ✱ t❤❡r❡ ❡①✐sts y ❛♥❞ z ∈ M - ∞ s✉❝❤ t❤❛t x = y + z ❛♥❞ ❢♦r ❛•• p > 1✿ R(dy) p + C(dz) p ≤ k p x p . Pr♦♦❢✳ ❇② t❤❡♦r❡♠ 3.2.1 ❛♣♣•✐❡❞ ❢♦r p = 1✱ t❤❡r❡ ❡①✐sts y ′ ❛♥❞ z ′ ∈ S(M) s✉❝❤ t❤❛t dx = y ′ + z ′ ❛♥❞ ❢♦r ❛•• t ≥ 0✱ K t (Ry ′ , 1, ∞) ≤ C 1 K t (Gx, 1, ∞) ❛♥❞ K t (Cz ′ , 1, ∞) ≤ C 1 K t (Gx, 1, ∞)✳ ❇② r❡❛• ✐♥t❡r♣♦•❛t✐♦♥✱ t❤✐s ♠❡❛♥s t❤❛t ❢♦r ❛•• p > 1✱ Ry ′ p + Cz ′ p ≤ 2C 1 Gx p ✳ ❉❡✜♥❡ ∆ 0 = E 0 ❛♥❞ ❢♦r n ≥ 1✱ ∆ n = E n -E n-1 ✳ ▲❡t dy = (∆ n (y ′ n )) n≥0 ❛♥❞ dz = (∆ n (z ′ n )) n≥0 ✳ ❚❤✐s ✇❛② y ❛♥❞ z ❜❡•♦♥❣ t♦ M - ∞ ❛♥❞ t❤❡② ❦❡❡♣ t❤❡ s❛♠❡ ♣r♦♣❡rt✐❡s✳ ■♥❞❡❡❞✱ ❢♦r ❛•• n ∈ N✱ dx n = ∆ n (dx n ) = ∆ n (y ′ n + z ′ n ) = dy n + dz n ❛♥❞ ❜② ❙t❡✐♥✬s ✐♥❡q✉❛•✐t② ✭❬✹✾(✱ ❬✻✷(✮✿ R(dy) p + C(dz) p Ry ′ p + Cz ′ p G(dx) p . ▼♦r❡♦✈❡r✱ ❜② t❤❡ t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛•✐t② ♦❢ ♠❛rt✐♥❣❛• ❞✐✛❡r❡♥❝❡s ❬✹✾(✱ G(dx) p ≈ x p ✳ ❍❡♥❝❡✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t k p ✭✐♥❞❡♣❡♥❞❛♥t ♦❢ x✮ s✉❝❤ t❤❛t✿ R(dy) p + C(dz) p ≤ k p x p . ✸✳✺✳ ❖◆ ❚❍❊ P|❖P❊|❚■❊❙ KH + ❆◆❉ KH - ✼✼ |❡♠❛r❦ ✸✳✹✳✷✳ ❙✐♥❝❡ ❜② ✐♥t❡r♣♦•❛t✐♦♥✱ ❇✉r❦❤♦•❡r✲•✉♥❞② ✐♥❡q✉❛•✐t② st❛②s tr✉❡ ✐♥ ❛•• ✐♥t❡r♣♦•❛t✐♦♥ s♣❛❝❡ E ❜❡t✇❡❡♥ L p ✲s♣❛❝❡s✱ ∞ > p > 1✱ t❤❡ ❛r❣✉♠❡♥t ❛❜♦✈❡ ❝❛♥ ❜❡ r❡♣r♦❞✉❝❡❞ ❛♥❞ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ y, z ❝♦♥str✉❝t❡❞ ✐♥ t❤❡ ♣r♦♦❢ ✈❡r✐✜❡s Ry E + Cz E ≤ k E x E . ✸✳✺ ❖♥ t❤❡ ♣r♦♣❡rt✐❡s Kh + ❛♥❞ Kh - ✸✳✺✳✶ ❊①♣•✐❝✐t ❝♦✉♥t❡r✲❡①❛♠♣•❡s ✐♥ L 2,∞ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❝♦♥str✉❝t t✇♦ ❝♦✉♥t❡r✲❡①❛♠♣•❡s t♦ ♣r♦✈❡ ♣r♦♣♦s✐t✐♦♥ ✸✳✶✳✹✳ ✸✳✺✳✷ ❈♦✉♥t❡r✲❡①❛♠♣•❡ ❢♦r H 2,∞ = R 2,∞ + C 2,∞ ❚❤❡ ✜rst ❝♦✉♥t❡r✲❡①❛♠♣•❡ ✐s ✈❡r② s✐♠♣•❡ t♦ ❞❡✜♥❡✳ ❋✐① N ∈ N✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❝♦♥str✉❝t ❛ s❡q✉❡♥❝❡ ♦❢ ♠❛tr✐❝❡s x = (x i ) 1≤i≤N s✉❝❤ t❤❛t Gx 2,∞ ≈ √ ln N ❛♥❞ Rx 2,∞ = 1✳ ■♥ ❝♦♥s❡q✉❡♥❝❡✱ ✇❡ ❣❡t✿ x R2,∞+C2,∞ ≤ Rx 2,∞ ≤ 1 √ ln N Gx 2,∞ . ▲❡♠♠❛ ✸✳✺✳✶✳ ▲❡t x = (x i ) 1≤i≤N ∈ B(H) ✇✐t❤ x i = 1 √ i e 1,i ❢♦r 1 ≤ i ≤ N ✳ ❚❤❡♥✱ Gx 2,∞ ≥ ln(N -1) ❛♥❞ Cx 2,∞ = 1✳ Pr♦♦❢✳ ❲❡ ❤❛✈❡✿ |Gx * | 2 = N i=1 1 i e 1,1 ⊗ 1 A ≥ ln(N -1)e 1,1 ⊗ 1 A ❛♥❞✿ |Cx| 2 = N i=1 1 i e i,i . ❍❡♥❝❡ |Gx * | 2 1,∞ ≥ ln(N -1) ❛♥❞ |Cx| 2 1,∞ ❂ ✶✳ ✸✳✺✳✸ ❈♦✉♥t❡r✲❡①❛♠♣•❡ ❢♦r H 2,∞ = R 2,∞ ∩ C 2,∞ ■♥ t❤✐s s✉❜s❡❝t✐♦♥✱ ✇❡ ✇✐•• ❛•✇❛②s ❛ss✉♠❡ t❤❛t t❤❡ (ξ i ) i≥0 ❛r❡ ❢r❡❡ ❍❛❛r ✉♥✐t❛r✐❡s✳ ❲❡ st❛rt ❜② ♠❛❦✐♥❣ t❤❡ ❢♦••♦✇✐♥❣ ♦❜s❡r✈❛t✐♦♥ t❤❛t ✐s ❡❛s✐•② ❞❡❞✉❝❡❞ ❢r♦♠ ❬✶✹(✿ ▲❡♠♠❛ ✸✳✺✳✷✳ ▲❡t x ∈ S(M)✳ ■❢ ❛•• ❡✐❣❡♥✈❛•✉❡s ♦❢ |Rx| ❛r❡ •❡ss ❛r❡ ❡q✉❛• t♦ t❤❡ ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛•✉❡s ♦❢ |Cx|✱ t❤❡♥ µ(Gx) ≤ 2µ(Cx). Pr♦♦❢✳ ❈♦♥s✐❞❡r t❤❡ ❡•❡♠❡♥t Rx ⊕ Cx ∈ (M ⊗ B(ℓ 2 )) ⊕ (M ⊗ B(ℓ 2 )) ✇❤✐❝❤ ✐s ♥♦ •♦♥❣❡r ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♣r♦❜❛❜✐•✐t② s♣❛❝❡ ❜✉t ✐s st✐•• ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ✐♥t❡❣r❛t✐♦♥ s♣❛❝❡ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ s✉♠ tr❛❝❡✳ ■♥ ❬✶✹( ✭♣✳3✮✱ ✐t ✐s ♣r♦✈❡❞ t❤❛t λ 2t (Gx) ≤ λ t (Rx ⊕ Cx) ❢♦r ❛•• t ≥ 0✳ ❙②♠♠❡tr✐❝❛••②✱ ❢♦r ❛•• t ≥ 0✱ µ t (Gx) ≤ 2µ t (Rx ⊕ Cx) ✇❤✐❝❤ ✐s t❤❡ ❦❡② ✐♥❡q✉❛•✐t② ✇❡ ✇✐•• ♥❡❡❞✳ ❋✉rt❤❡r♠♦r❡✱ r❡❝❛•• t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥♥❛• ❡①♣❡❝t❛t✐♦♥ Id ⊗ τ s❡♥❞s |Gx| 2 t♦ |Cx| 2 ✐♥ M ⊗ A✳ ❚❤✐s t❤❛t ❢♦r ❛•• t ≥ 0✿ t 0 µ(Cx) 2 ≤ t 0 µ(Gx) 2 . ■♥ ♣❛rt✐❝✉•❛r✱ ❢♦r t 0 = τ (s✉♣♣ |Cx|)✱ ✇❡ ♦❜t❛✐♥✿ τ (|Cx| 2 ) = t0 0 µ(Cx) 2 ≤ t0 0 µ(Gx) 2 ≤ ∞ 0 µ(Gx) 2 = τ (|Gx| 2 ) = τ (|Cx| 2 ). ❚❤❡r❡❢♦r❡✱ ❛•• t❡r♠s ❛♣♣❡❛r✐♥❣ ❛❜♦✈❡ ❛r❡ ❡q✉❛• ❛♥❞ µ t (Gx) = 0 ❢♦r ❛•• t > t 0 ✳ ❍❡♥❝❡✱ µ t (Gx) ≤ 1 t≤t0 2µ t (Rx ⊕ Cx)✳ ❋✐♥❛••②✱ ❜② ❤②♣♦t❤❡s✐s✱ ❛•• ❡✐❣❡♥✈❛•✉❡s ♦❢ |Rx| ❛r❡ •❡ss ❛r❡ ❡q✉❛• t♦ t❤❡ ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛•✉❡s ♦❢ |Cx|✳ ■♥ t❤✐s s♣❡❝✐❛• ❝❛s❡✱ µ(Rx ⊕ Cx) ✐s ✈❡r② ❡❛s② t♦ ❝♦♠♣✉t❡✳ ■♥❞❡❡❞✱ ❝♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥ f ❞❡✜♥❡❞ ❜②✿ f = µ t (Cx) ✐❢ t ≤ t 0 µ t-t0 (Rx) ✐❢ t > t 0 . ❖♥❡ ❝❛♥ ❡❛s✐•② s❡❡ t❤❛t f ❤❛s t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥ ❛s Rx ⊕ Cx✱ f ✐s ♥♦♥✐♥✲ ❝r❡❛s✐♥❣ ❛♥❞ •❡❢t✲❝♦♥t✐♥✉♦✉s s♦ f = µ(Rx ⊕ Cx)✳ ■♥ ❝♦♥s❡q✉❡♥❝❡✱ ❢♦r ❛•• t ≥ 0✿ µ t (Gx) ≤ 1 t≤t0 2f (t) = 2µ t (Cx). ❯s✐♥❣ t❤❡ ♣r❡✈✐♦✉s •❡♠♠❛✱ ✇❡ ✇✐•• ❜❡ ❛❜•❡ t♦ ❝♦♥tr♦• t❤❡ ♥♦r♠ ♦❢ Gx✳ ❲❤❛t ✐s •❡❢t t♦ ❞♦ ♥♦✇ ✐s t♦ ✜♥❞ ❛ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t t❤❡ ❡✐❣❡♥✈❛•✉❡s ♦❢ Cx ❞♦♠✐♥❛t❡ t❤♦s❡ ♦❢ Rx ❛♥❞ Cx 2,∞ ≪ Rx 2,∞ ✳ ■♥ t❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤❡ s❡❝t✐♦♥✱ ✇❡ ❝♦♥str✉❝t ❛♥ ❡①♣•✐❝✐t ❡①❛♠♣•❡ ✇✐t❤ ♦♥❝❡ ❛❣❛✐♥ M = B(ℓ 2 )✳ ❋✐① N ∈ N✳ ❈♦♥s✐❞❡r t❤❡ s❡q✉❡♥❝❡✿ u n = n k=1 k( N k - N k + 1 ) ❛♥❞ ❞❡✜♥❡ n j = inf{n : u n ≥ j} ✇✐t❤ t❤❡ ❝♦♥✈❡♥t✐♦♥ n j = ∞ ✐❢ j > sup n≥1 u n = u N ✳ ▲❡♠♠❛ ✸✳✺✳✸✳ ❋♦r ❛•• i✱ j≥1 1 i≤nj 1 n j = N i . ❆♥❞ ❢♦r ❛•• j ≥ 1✱ i≥1 1 i≤nj 1 n j = 1 j≤u N . ✸✳✺✳ ❖◆ ❚❍❊ P|❖P❊|❚■❊❙ KH + ❆◆❉ KH - ✼✾ Pr♦♦❢✳ ❇② ❞❡✜♥✐t✐♦♥ ♦❢ n j ✱ (i ≤ n j ) ⇔ (j > u i-1 )✳ ❍❡♥❝❡✱ j>0 1 i≤nj 1 n j = j>ui-1 1 n j = k≥i u k j=u k-1 +1 1 n j = k≥i k( N k - N k + 1 ) 1 k = N i . ❚❤❡ ♦t❤❡r ❡q✉❛•✐t② ✐s ❞✐r❡❝t ✇✐t❤ t❤❡ ❝♦♥✈❡♥t✐♦♥ 1 ∞ = 0✳ ▲❡t x i,j = 1 i≤nj √ n j e i,j ∈ B(ℓ 2 ) ❢♦r i, j ≥ 1✳ Pr♦♣♦s✐t✐♦♥ ✸✳✺✳✹✳ ▲❡t (ξ i,j ) i,j ❜❡ ❢r❡❡ ❍❛❛r ✉♥✐t❛r✐❡s✳ ❈♦♥s✐❞❡r t❤❡ x i,j ❞❡✲ ✜♥❡❞ ❛❜♦✈❡✳ ❲❡ ❤❛✈❡✿ i,j x i,j ⊗ ξ i,j 2,∞ ≤ 2 √ N , ❛♥❞✿ i,j |x i,j | 2 ≈ N ln(N ).
Pr♦♦❢✳ ❯s✐♥❣ t❤❡ ♣r❡✈✐♦✉s •❡♠♠❛✱ ✇❡ ❤❛✈❡✿ i,j

x * i,j

2 = i,j≥1 1 i≤nj 1 n j e i,i = i≥1 N i e i,i . ❈♦♥s❡q✉❡♥t•②✱ ❢♦r ❛•• t ≥ 0✱ µ t ( i,j x * i,j 2 ) = N ⌊t + 1⌋ ≤ 1 t . ❙✐♠✐•❛r•②✱ i,j |x i,j | 2 = i,j≥1 1 i≤nj 1 n j e j,j = j≥1 1 j≤u N e j,j . ❙♦ ✇❡ ❤❛✈❡ µ( i,j |x i,j | 2 ) = 1 0,u N . ❚❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ •❡♠♠❛ ✸✳✺✳✷ ❛r❡ s❛t✐s✜❡❞ s♦✿ µ( i,j x i,j ⊗ ξ i,j ) ≤ 2µ(( i,j x * i,j 2 ) 1/2 ). ❍❡♥❝❡✱ i,j x i,j ⊗ ξ i,j 2,∞ ≤ 2 i,j x * i,j 2 1,∞ ≤ 2 √ N ✳ ❋✉rt❤❡r♠♦r❡✱ ♥♦t❡ t❤❛t✿ u N = ∞ k=1 k( N k - N k + 1 ) = ∞ k=1 N k ≈ N ln(N ). ❙♦✱ ❛s ❡①♣❡❝t❡❞✱ ✇❡ ❤❛✈❡ i,j |x i,j | 2 1,∞ ≈ N ln(N )✳ ✽✵ ❈❍❆P❚❊| ✸✳ ◆❖◆❈❖▼▼❯❚❆❚■❱❊ ❑❍■◆❚❈❍■◆❊ ■◆❊◗❯❆▲■❚■❊❙ ✸✳✺✳✹ ❚❤❡ s②st❡♠❛t✐❝ ❝♦♥str✉❝t✐♦♥ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❣✐✈❡ ❛ s❡❝♦♥❞ ♣r♦♦❢ ♦❢ ♣r♦♣♦s✐t✐♦♥ ✸✳✶✳✹ ❛♥❞ ♠♦r❡ ✐♠♣♦rt❛♥t•②✱ ✇❡ ♣r♦✈❡ t❤❡♦r❡♠ ✸✳✶✳✺✳ ❚❤❡ ♣r♦♦❢s ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ❙❝❤✉r✲❍♦r♥ t❤❡♦r❡♠ ✇❤✐❝❤ ❡♥❛❜•❡s ✉s t♦ s❤♦✇ t❤❛t t❤❡ ❞✐str✐❜✉t✐♦♥s ♦❢ Gx ❛♥❞ Rx ❝❛♥ ❜❡ ♣r❡s❝r✐❜❡❞✳ ❋♦r ♥♦✇✱ t❤❡ ξ i ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ ♦rt❤♦♥♦r♠❛•✳ ❆ ✜♥✐t❡ s❡q✉❡♥❝❡ a = (a 1 , ..., a n ) ✇✐•• ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ a = (a 1 , ..., a n , 0, 0, 0, ...)✳ ❚♦ ❛♥② s❡q✉❡♥❝❡ a ✇❡ ❛ss♦❝✐❛t❡ t❤❡ ❢✉♥❝t✐♦♥ f a := ∞ i=1 a i 1 (i-1,i] . Pr♦♣♦s✐t✐♦♥ ✸✳✺✳✺✳ ▲❡t M = B(ℓ 2 )✳ ▲❡t a ❛♥❞ b ❜❡ t✇♦ ✜♥✐t❡ ♥♦♥✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡s ♦❢ ♣♦s✐t✐✈❡ r❡❛•s s✉❝❤ t❤❛t ❢♦r ❛•• n ∈ N✱ n i=1 a 2 i ≥ n i=1 b 2 i ❛♥❞ ∞ i=1 a 2 i = ∞ i=1 b 2 i .
❚❤❡♥✱ t❤❡r❡ ❡①✐sts x ∈ S(M) s✉❝❤ t❤❛t

µ(Gx) = µ(Cx) = f a ❛♥❞ µ(Rx) = f b ✳ Pr♦♦❢✳ ▲❡t N ∈ N ❜❡ t❤❡ •❡♥❣t❤ ♦❢ a ❛♥❞ b✳ ❚❤❡ ❙❝❤✉r✲❍♦r♥ t❤❡♦r❡♠ ❛♣♣•✐❡❞ t♦ (a 2 i ) ❛♥❞ (b 2 i ) ♣r♦❞✉❝❡s ❛ ♣♦s✐t✐✈❡ ♠❛tr✐① M ∈ M N (C) s✉❝❤ t❤❛t t❤❡ ❡✐❣❡♥✈❛•✉❡s ♦❢ M ❛r❡ ❣✐✈❡♥ ❜② (a 2 i ) ❛♥❞ t❤❡ ❞✐❛❣♦♥❛• ♦❢ M ✐s ❣✐✈❡♥ ❜② (b 2 i )✳ ❈♦♥s✐❞❡r x = (e i,i M 1/2 ) 0≤i≤N ✳ ❚❤❡♥ |Gx| 2 = i,j M 1/2 e i,i e j,j M 1/2 ⊗ ξ * i ξ j = i M 1/2 e i,i M 1/2 ⊗ 1 = M ⊗ 1. ❙✐♠✐•❛r ❝♦♠♣✉t❛t✐♦♥s ❣✐✈❡✿ |Cx| 2 = M ⊗e 1,1 ❛♥❞ |(Rx) * | 2 = Diag(M )⊗e 1,1 ✳ ❙❡❝♦♥❞ ♣r♦♦❢ ♦❢ ♣r♦♣♦s✐t✐♦♥ ✸✳✶✳✹✳ ❋✐① N ∈ N✳ ❉❡♥♦t❡ ❜② u N t❤❡ q✉❛♥t✐t② u N := N i=1 1/i✳ ▲❡t a = (a i ) i≤1 ❛♥❞ b = (b i ) i≤N ❜❡ ❞❡✜♥❡❞ ❜② a 1 = √ u N ✱ a i = 0 ❢♦r i ≥ 2✱ b i = 1/i ❢♦r i ≤ N ❛♥❞ b i = 0 ❢♦r i > N ✳ ❆♣✲ ♣•②✐♥❣ t❤❡ ♣r❡✈✐♦✉s ♣r♦♣♦s✐t✐♦♥✱ ✇❡ ♦❜t❛✐♥ ❛♥ ❡•❡♠❡♥t x ∈ S(M) s✉❝❤ t❤❛t Gx 2,∞ = √ u N ≈ ln(N ) ❛♥❞ Rx 2,∞ = 1✳ ❍❡♥❝❡✱ ✇❡ ❝❛♥♥♦t ❤❛✈❡ . H2,∞ . R2,∞+C2,∞ ✳ ◆♦✇ ❞❡✜♥❡ v N := N i=1 ⌊N/i⌋✳ ▲❡t a = ( ⌊N/i⌋) i≤N ❛♥❞ b = (1) i≤v N ✳ ❇② ❛♣♣•②✐♥❣✱ t❤❡ ♣r❡✈✐♦✉s ♣r♦♣♦s✐t✐♦♥ ❛❣❛✐♥✱ ✇❡ ♦❜t❛✐♥ ❛♥ ❡•❡♠❡♥t x ∈ S(M) s✉❝❤ t❤❛t Gx 2,∞ = √ N ❛♥❞ Rx 2,∞ = √ v N ≈ N ln(N ) ✇❤✐❝❤ ❞❡♥✐❡s t❤❡ ♣♦ss✐❜✐•✐t② ♦❢ ❤❛✈✐♥❣ . H2,∞ . R2,∞∩C2,∞ ✳ ▲❡t ✉s ♥♦✇ ♣r♦✈❡ t❤❡♦r❡♠ ✸✳✶✳✺✳ ❲❡ ❞✐✈✐❞❡ t❤❡ ♣r♦♦❢ ✐♥t♦ t❤r❡❡ •❡♠♠❛s✳ |❡❝❛•• t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ s❡t F ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ ❞②❛❞✐❝ st❡♣ ❢✉♥❝t✐♦♥s✿ F := {f ∈ L ∞ (0, ∞) : ∃n ∈ N, ∃N ∈ N, ∃a ∈ (R + ) N , f = N i=1 a i 1 [(i-1)2 -n ,i2 -n ) }. ▲❡♠♠❛ ✸✳✺✳✻✳ ▲❡t M = B(ℓ 2 )⊗L ∞ (0, 1)✳ ▲❡t f, g ∈ F s✉❝❤ t❤❛t f 2 ≻ g 2 ❛♥❞ f 2 = g 2 t❤❡♥ t❤❡r❡ ❡①✐sts x ∈ S(M) s✉❝❤ t❤❛t µ(Gx) = f ❛♥❞ µ(Rx) = g✳ Pr♦♦❢✳ ❚❤✐s ✐s ❛ ❞✐r❡❝t ❝♦r♦••❛r② ♦❢ ♣r♦♣♦s✐t✐♦♥ ✸✳✺✳✺✳ ❲r✐t❡ f = N i=1 a i 1 [(i-1)2 -n ,i2 -n ) ❛♥❞ g = N i=1 b i 1 [(i-1)2 -n ,i2 -n ) ✳ ❙✐♥❝❡ f 2 ≻ g 2 ✱ a 2 ≻ b 2 ✳ ▲❡t x S(B(ℓ 2 )) ❜❡ ❣✐✈❡♥ ❜② ♣r♦♣♦s✐t✐♦♥ ✸✳✺✳✺ ❛♥❞ ❞❡✜♥❡ x ′ = (x i ⊗ 1 (0,2 -n ) ) i≤N ✳ ❚❤❡♥ µ(Gx ′ ) = f ❛♥❞ µ(Rx ′ ) = g✳ ▲❡♠♠❛ ✸✳✺✳✼✳ ▲❡t E ❜❡ ❛ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡ ❛♥❞ M = B(ℓ 2 )⊗L ∞ (0, 1)✳ ❚❤❡♥✱ Kh + (E, M) ⇒ ∀f, g ∈ F, f 2 ≻ g 2 ⇒ g E f E , ❛♥❞ Kh -(E, M) ⇒ ∀f, g ∈ F, f 2 ⊲ g 2 ⇒ g E f E . Pr♦♦❢✳ ❙✉♣♣♦s❡ Kh + (E, M)✳ ▲❡t f, g ∈ F s✉❝❤ t❤❛t f 2 ≻ g 2 ✳ ❲❡ ❝❛♥ ✜♥❞ h ≥ g ∈ F s✉❝❤ t❤❛t f 2 ≻ h 2 ❛♥❞ f 2 = h 2 ❜② •❡♠♠❛ ✶✳✹✳✷✳ ❚❤❡♥ ❜② •❡♠♠❛ ✸✳✺✳✻ t❤❡r❡ ❡①✐sts x ∈ S(M) s✉❝❤ t❤❛t µ(Gx) = f ❛♥❞ µ(Rx) = h✳ ❍❡♥❝❡✱ ❜② Kh + (E, M)✱ f E = Gx E Rx E = h E . ❙✉♣♣♦s❡ Kh -(E, M)✳ ▲❡t f, g ∈ F s✉❝❤ t❤❛t f 2 ⊲ g 2 ✳ ❲❡ ❝❛♥ ✜♥❞ h ≥ g ∈ F s✉❝❤ t❤❛t f 2 ⊲ h 2 ❛♥❞ h 2 = f 2 ❛❣❛✐♥ ❜② •❡♠♠❛ ✶✳✹✳✷✳ ❙♦ ❜② •❡♠♠❛ ✸✳✺✳✻✱ t❤❡r❡ ❡①✐sts x ∈ S(M) s✉❝❤ t❤❛t µ(Gx) = h ❛♥❞ µ(Rx) = f ✳ ❇② Kh -(E, M)✱ Gx E x R E +C E ✳ ■♥ ♣❛rt✐❝✉•❛r✱ Gx E Rx E ✳ ❍❡♥❝❡✱ h E f E ✳ |❡❝❛•• t❤❛t g ≤ h s♦ g E ≤ h E ❛♥❞ t❤❡ ♣r♦♦❢ ✐s ❝♦♠♣•❡t❡✳ ▲❡♠♠❛ ✸✳✺✳✽✳ ▲❡t E ❜❡ ❛ s②♠♠❡tr✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡✳ ■❢ E ✐s •❡❢t✲2✲♠♦♥♦t♦♥❡ t❤❡♥ Kh + (E, M) ❤♦•❞s✳ ■❢ ❢✉rt❤❡r♠♦r❡✱ t❤❡r❡ ❡①✐sts p > 0 s✉❝❤ t❤❛t E ∈ Int(L p , L ∞ )✱ E ✐s r✐❣❤t✲2✲♠♦♥♦t♦♥❡ t❤❡♥ Kh -(E, M) ❤♦•❞s✳ Pr♦♦❢✳ ❈♦♥s✐❞❡r t❤❡ ❝♦♥❞✐t✐♦♥❛• ❡①♣❡❝t❛t✐♦♥ E : M A → M ❞❡✜♥❡❞ ❜② E := Id ⊗ τ A ✳ ◆♦t❡ t❤❛t ❢♦r ❛♥② x ∈ S(M)✱ E(|Gx| 2 ) = |Cx| 2 ❛♥❞ E(|(Gx) * | 2 ) = |(Rx) * | 2 . ✭✸✳✺✮ ❙✉♣♣♦s❡ t❤❛t ♥♦✇ t❤❛t E ✐s •❡❢t✲2✲♠♦♥♦t♦♥❡✳ ❇② ✭✸✳✺✮✱ |Cx| 2 ≺ |Gx| 2 ❛♥❞ |Rx| 2 ≺ |Gx| 2 s♦ Gx E ≥ Cx E ❛♥❞ Gx E ≥ Rx E ✳ |❡❝✐♣r♦❝❛••②✱ ❜② r❡♠❛r❦ ✸✳✸✳✻ ♦r ❬✶✹(✱ Gx E x R E ∩C E ✳ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ❡①✐sts p > 0 s✉❝❤ t❤❛t E ∈ Int(L p , L ∞ ) ❛♥❞ t❤❛t E ✐s r✐❣❤t✲2✲♠♦♥♦t♦♥❡✳ ❋✐① x ∈ S(M)✱ t❤❡♥ ❜② t❤❡♦r❡♠ ✸✳✶✳✽✱ x R E +C E Gx E ✳ |❡❝✐♣r♦❝❛••②✱ ❜② ✭✸✳✺✮✱ |Cy| 2 ⊲ |Gy| 2 ❛♥❞ |Ry| 2 ⊲ |Gy| 2 ❢♦r ❛•• y ∈ S(M)✳ ◆♦✇ ✜① y, z ∈ S(M) s✉❝❤ t❤❛t y + z = x✳ ❯s✐♥❣ r✐❣❤t✲2✲♠♦♥♦t♦♥✐❝✐t②✱ Gx E Gy E + Gz E ≤ Ry E + Cz E . ❙♦ E ❤❛s Kh -(E, M)✳ Pr♦♦❢ ♦❢ t❤❡♦r❡♠ ✸✳✶✳✺✳ ❚❤❡ ✐♠♣•✐❝❛t✐♦♥s ❢r♦♠ r✐❣❤t t♦ •❡❢t ❛r❡ ❣✐✈❡♥ ❜② •❡♠♠❛ ✸✳✺✳✽✳ ❚❤❡ ✐♠♣•✐❝❛t✐♦♥s ❢r♦♠ •❡❢t t♦ r✐❣❤t ❛r❡ ♦❜t❛✐♥❡❞ ❜② •❡♠♠❛ ✸✳✺✳✼ ❝♦♠❜✐♥❡❞ ✇✐t❤ t❤❡ ❋❛t♦✉ ♣r♦♣❡rt② ❛♥❞ ✐♥ ♣❛rt✐❝✉•❛r ✐ts ❝♦♥s❡q✉❡♥❝❡s ❣✐✈❡♥ ❜② •❡♠♠❛s ✶✳✹✳✺ ❛♥❞ ✶✳✹✳✽✳ ❈❤❛♣t❡r ✹ ❲❡❛❦ t②♣❡ ✐♥❡q✉❛•✐t② ❢♦r s✐♥❣✉•❛r ✐♥t❡❣r❛•s ✹✳✶ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♠❛✐♥ r❡s✉•t ♦❢ t❤✐s ❝❤❛♣t❡r ❜❡•♦♥❣s t♦ t❤❡ ♥♦✇ ✇❡•• ❞❡✈❡•♦♣♣❡❞ t❤❡♦r② ♦❢ s✐♥❣✉•❛r ✐♥t❡❣r❛•s✳ ❚❤❡ •❛tt❡r ✇❛s ✐♥✐t✐❛t❡❞ ✐♥ t❤❡ ✶✾✺✵✬s ❜② ❈❛•❞❡ró♥ ❛♥❞ )②❣♠✉♥❞ ✇❤♦ ❢♦✉♥❞ ❛ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ❛ ❦❡r♥❡• ♦♣❡r❛t♦r t♦ ❜❡ ❜♦✉♥❞❡❞ ♦♥ L p ❢♦r 1 < p < ∞✳ ■t ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✭✇✐t❤♦✉t ❞❡t❛✐•s ❛❜♦✉t ❞❡✜♥✐t✐♦♥✮ ❜② t❤❡ ❢♦••♦✇✐♥❣ t❤❡♦r❡♠✿ ❚❤❡♦r❡♠ ✹✳✶✳✶✳ ▲❡t n ∈ N✱ p ∈ (1, ∞) ❛♥❞ k ❛ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥ ❢r♦♠ R 2n t♦ C ✈❡r✐❢②✐♥❣ t❤❡ s✐③❡ ❛♥❞ s♠♦♦t❤♥❡ss ❝♦♥❞✐t✐♦♥s✳ ❚❤❡♥ ✐❢ t❤❡ ❢♦r♠❛• ❡①♣r❡ss✐♦♥ ✿ T f (x) = y∈R n k(x, y)f (y)dy ❞❡✜♥❡s ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r T ♦♥ L 2 (R n )✱ ✐t ❛•s♦ ❞❡✜♥❡s ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r ✭st✐•• ♥♦t❡❞ T ✮ ♦♥ L p (R n )✳ T ✐s ❝❛••❡❞ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r ❛♥❞ k ✐ts ❦❡r♥❡•✳ ❙✐③❡ ❛♥❞ s♠♦♦t❤♥❡ss ✇✐•• ❜❡ ❞❡✜♥❡❞ •❛t❡r ✐♥ ❛ ♠♦r❡ ❣❡♥❡r❛• ❝♦♥t❡①t✳ ❋♦r t❤❡ p = 1 ❝❛s❡✱ ♦♥•② ✇❡❛❦ ❜♦✉♥❞❡❞♥❡ss ✐s tr✉❡ ✐♥ ❣❡♥❡r❛•✿ ❚❤❡♦r❡♠ ✹✳✶✳✷✳ ❲✐t❤ t❤❡ s❛♠❡ ❝♦♥❞✐t✐♦♥s ❛♥❞ ♥♦t❛t✐♦♥s✱ T ❞❡✜♥❡s ❛♥ ♦♣❡r❛t♦r ♦♥ L 1 (R n ) ❛♥❞ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C s✉❝❤ t❤❛t ❢♦r ❛•• f ∈ L 1 (R n )✿ sup t>0 tµ{T f > t} ≤ C f 1 ✇❤❡r❡ µ ✐s t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳ ❆ ♠♦t✐✈❛t✐♦♥ t♦ s❤♦✇ t❤✐s s❡❝♦♥❞ t❤❡♦r❡♠ ✐s t❤❛t ✐t ❞✐r❡❝t•② ✐♠♣•✐❡s t❤❡ ✜rst ♦♥❡ ❜② r❡❛• ✐♥t❡r♣♦•❛t✐♦♥ ❛♥❞ ❞✉❛•✐t②✳ ❚❤❡ ✐❞❡❛s ❜❡❤✐♥❞ t❤❡s❡ t✇♦ t❤❡♦r❡♠s r❡♠❛✐♥ ✈❛•✐❞ ❢♦r ❦❡r♥❡•s ❛♥❞ ❢✉♥❝✲ t✐♦♥s t❛❦✐♥❣ ✈❛•✉❡s ✐♥ ❞✐✛❡r❡♥t ✈❡❝t♦r s♣❛❝❡s✱ ✇❤✐❝❤ ♠❛❦❡s t❤❡♠ ❛ ❣r❡❛t t♦♦• ✽✸ ✽✹❈❍❆P❚❊| ✹✳ ❲❊❆❑ ❚(P❊ ■◆❊◗❯❆▲■❚( ❋❖| ❙■◆•❯▲❆| ■◆❚❊•|❆▲❙ t♦ s❤♦✇ ❜♦✉♥❞❡❞♥❡ss ♦❢ ❝❡rt❛✐♥ ♦♣❡r❛t♦rs s✉❝❤ ❛s ❣❡♥❡r❛•✐s❡❞ ❍✐•❜❡rt tr❛♥s❢♦r♠ ♦r ▲✐tt•❡✇♦♦❞✲P❛•❡② ✐♥❡q✉❛•✐t✐❡s✳ ❲❤❛t ✇❡ ♣r♦✈❡ ✐♥ t❤✐s ❝❤❛♣t❡r ✐s ❛ ❣❡♥❡r❛•✐✲ s❛t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ t❤❡♦r❡♠ t♦ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ✐♥t❡❣r❛t✐♦♥✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ r❡s✉•t ❤❛s ❛•r❡❛❞② ♣r♦✈❡♥ t♦ ❜❡ ✉s❡❢✉• s✐♥❝❡ ✐t ✐s t❤❡ ♠❛✐♥ ✐♥❣r❡❞✐❡♥t ✉s❡❞ ✐♥ ❬✼( ✐♥ ✇❤✐❝❤ ✐t ✐s s❤♦✇♥ t❤❛t t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c s✉❝❤ t❤❛t ❢♦r ❛♥② ▲✐♣s❤✐t③ ❢✉♥❝t✐♦♥ f ❛♥❞ ❢♦r ❛♥② s❡•❢✲❛❞•♦✐♥ts ♦♣❡r❛t♦rs x ❛♥❞ y✱ f (x) -f (y) 1,∞ ≤ c f ′ ∞ x -y 1 . ❚❤✐s ✐♥❡q✉❛•✐t② ❞♦❡s ♥♦t ❞✐r❡❝t•② ❡①♣r❡ss t❤❡ ✇❡❛❦ (1, 1) ❜♦✉♥❞❡❞♥❡ss ♦❢ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r ❜✉t ✐s r❡❞✉❝❡❞ t♦ ✐t ✐♥ t❤❡ ♠❡♥t✐♦♥♥❡❞ ♣❛♣❡rs✳ ❲❡ ❛•s♦ ❤♦♣❡ t❤❛t ♦✉r ♠❛✐♥ r❡s✉•t ✐s ❛ ✇❛② t♦ t❛❝❦•❡ ❣❡♥❡r❛•✐s❛t✐♦♥s ♦❢ ❝•❛ss✐❝❛• ✐♥❡q✉❛•✐t✐❡s ♦♥ L p ✇❤♦s❡ ♣r♦♦❢ r❡•✐❡s ♦♥ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ t❤❡♦r②✳ ❆♥♦t❤❡r ❛♣♣r♦❛❝❤ ❢♦r t❤✐s ❦✐♥❞ ♦❢ ♣r♦❜•❡♠ ✐s t♦ s❤♦✇ ❇▼❖ ❜♦✉♥❞❡❞♥❡ss r❛t❤❡r t❤❛♥ ✇❡❛❦ ❜♦✉♥❞❡❞♥❡ss ❛♥❞ ❝♦♥❝•✉❞❡ t❤❛♥❦s t♦ ✐♥t❡r♣♦•❛t✐♦♥ t❤❡♦r②✳ ❚❤✐s str❛t✲ ❡❣② ✐s ♦❢t❡♥ ❡❛s✐❡r✳ ■t ✜rst ❛♣♣❡❛rs ✐♥ ❬✷✺( t♦ s❤♦✇ t❤❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ s♦♠❡ ❋♦✉r✐❡r ♠✉•t✐♣•✐❡rs✳ ❆♥❞ ❤❛s ❛•s♦ ❜❡❡♥ ❛♣♣•✐❡❞ ✐♥ ❬✻✶(✱ ✇❤❡r❡ L p ✲❜♦✉♥❞❡❞♥❡ss ♦❢ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦rs ✇✐t❤ ♦♣❡r❛t♦r ✈❛•✉❡❞ ❦❡r♥❡•s ❛♥❞ ❝♦•✉♠♥ ✈❛•✉❡❞ ❢✉♥❝t✐♦♥s ✐s ✉s❡❞ t♦ st✉❞② ❍❛r❞② s♣❛❝❡s ♦♥ q✉❛♥t✉♠ t♦r✐✳ ✹✳✶✳✶ ▼❛✐♥ t❤❡♦r❡♠ ❲❡ ✇✐•• ♥♦✇ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t❛t✐♦♥s t❤❛t ✇✐•• ❛••♦✇ ✉s t♦ st❛t❡ t❤❡ ♠❛✐♥ t❤❡♦r❡♠ ♦❢ t❤✐s ❝❤❛♣t❡r✳ ▲❡t ( M, τ M ) ❜❡ ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡ ❛♥❞ M ❛ ✈♦♥ ◆❡✉♠❛♥♥ s✉❜❛•❣❡❜r❛ ♦❢ M s✉❝❤ t❤❛t τ M r❡str✐❝t❡❞ t♦ M ✭❞❡♥♦t❡❞ τ M ✮ ✐s s❡♠✐✜♥✐t❡✳ L p (M) ✐s ♥❛t✉r❛••② ✐♥❝•✉❞❡❞ ✐♥ L p ( M) ❢♦r ❛•• 1 ≤ p ≤ ∞✳ ▲❡t M ′ ❜❡ t❤❡ ❝♦♠♠✉t❛♥t ♦❢ M✳ ❲❡ ✇✐•• ❝♦♥s✐❞❡r N ❛♥❞ N t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛•❣❡❜r❛s ♦❢ * ✲✇❡❛❦•② ♠❡❛s✉r❛❜•❡ M✲ ❛♥❞ M✲✈❛•✉❡❞ ❢✉♥❝t✐♦♥s ♦♥ R n ✐✳❡ t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ t❡♥s♦r ♣r♦❞✉❝ts M⊗L ∞ (R n ) ❛♥❞ M⊗L ∞ (R n )✱ τ ✇✐•• ❞❡♥♦t❡ t❤❡✐r ♥❛t✉r❛• tr❛❝❡ ✭✇✐t❤ ♥♦ ❛♠❜✐❣✉✐t② s✐♥❝❡ N ✐s ♥❛t✉r❛••② ✐♥❝•✉❞❡❞ ✐♥ N ✮✳ ▲❡t T ❜❡ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❦❡r♥❡• k : R n × R n → M ′ ∩ M✱ ❢♦r♠❛••② ❣✐✈❡♥ ❜② t❤❡ ❡①♣r❡ss✐♦♥✿ T f (x) = y∈R n k(x, y)f (y)dy ❉❡✜♥✐t✐♦♥ ✹✳✶✳✸✳ ❙❛② t❤❛t T ❤❛s ▲✐♣s❝❤✐t③ ♣❛r❛♠❡t❡r γ ✭0 < γ ≤ 1✮ ✐❢ ❢♦r ❛•• x✱y ❛♥❞ z ✐♥ R n ✈❡r✐❢②✐♥❣ |x -z| ≤ 1 2 |y -z|✱ t❤❡ ❢♦••♦✇✐♥❣ s♠♦♦t❤♥❡ss ❡st✐♠❛t❡s ❤♦•❞✿ k(x, y) -k(z, y) M ≤ |x -z| γ |y -z| n+γ , k(y, x) -k(y, z) M ≤ |x -z| γ |y -z| n+γ . ❲❡ ✇✐•• ❛•s♦ s✉♣♣♦s❡ t❤❛t T ✈❡r✐✜❡s t❤❡ s✐③❡ ❝♦♥❞✐t✐♦♥✱ ❢♦r ❛•• x, y ∈ R n ✿ ✹✳✶✳ ■◆❚|❖❉❯❈❚■❖◆ ✽✺ k(x, y) M ≤ 1 |x -y| n ✳ |❡♠❛r❦ ✹✳✶✳✹✳ ❚❤✐s s✐③❡ ❝♦♥❞✐t✐♦♥ ✇✐•• ♥❡✈❡r ❛♣♣❡❛r t❤r♦✉❣❤♦✉t t❤❡ ♣r♦♦❢✳ ■t ✐s ❤♦✇❡✈❡r ✉s❡❞ ✐♥ r❡♠❛r❦ ✹✳✶✳✻✳ ▼♦r❡♦✈❡r✱ ✐t ✐s ✐♠♣•✐❡❞ ❜② t❤❡ s♠♦♦t❤♥❡ss ❝♦♥❞✐t✐♦♥ ♣r♦✈✐❞❡❞ t❤❛t k ❣♦❡s t♦ 0 ❛t ✐♥✜♥✐t②✳ ❖♥ t❤❡ ❝♦♥tr❛r②✱ t❤❡ s♠♦♦t❤♥❡ss ❝♦♥❞✐t✐♦♥ ✇✐•• ❜❡ ❝r✉❝✐❛• t♦ ♠❛♥② ❝♦♠♣✉✲ t❛t✐♦♥s✳ ❆ ❍ör♠❛♥❞❡r t②♣❡ ❝♦♥❞✐t✐♦♥✿ sup x∈R n |x-y|>2|x-x ′ | |k(x, y) -k(x ′ , y)| dy < C ✇♦✉•❞ ♥♦t s✉✣❝❡ t♦ ✉s❡ t❤❡ ♣r♦♦❢ ✇❡ ♣r❡s❡♥t✳ ❲❡ ✇♦✉•❞ ❛t •❡❛st ♥❡❡❞ ❛ ❞❡❝❛② ♦❢ t❤❡ ❢♦••♦✇✐♥❣ t②♣❡✿ sup x∈R n |x-y|>α|x-x ′ | |k(x, y) -k(x ′ , y)| dy < C α n , α > 2. ❇✉t ✇❡ ❞♦ ♥♦t ❦♥♦✇ ✐❢ ✐t ✐s ❛ str♦♥❣ ❡♥♦✉❣❤ ❝♦♥❞✐t✐♦♥ s✐♥❝❡ ♣❛rt ♦❢ t❤❡ ♣r♦♦❢ ♦❢ ♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ r❡•✐❡s ♦♥ ♣♦✐♥t✇✐s❡ ❡st✐♠❛t❡s ♦❢ t❤❡ ❦❡r♥❡•✳ ❉❡✜♥❡✱ ❢♦r ❛•• t > 0 ❛♥❞ f ∈ N ✿ λ t (f ) = τ ({f > t})✳ ❚❤❡ ♠❛✐♥ t❤❡♦r❡♠ ✇❡ ❛r❡ ❛✐♠✐♥❣ t♦ ♣r♦✈❡ ✐♥ t❤✐s ❝❤❛♣t❡r ✐s t❤❡ ❢♦••♦✇✐♥❣✳ ❚❤❡♦r❡♠ ✹✳✶✳✺✳ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c n,γ ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ n ❛♥❞ γ s✉❝❤ t❤❛t ❢♦r ❛•• ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r T ✇✐t❤ (M ′ ∩ M)✲✈❛•✉❡❞ ❦❡r♥❡• ❛♥❞ ▲✐♣s❝❤✐t③ ♣❛r❛♠❡t❡r 0 < γ ≤ 1✱ ✈❡r✐❢②✐♥❣ t❤❡ s✐③❡ ❡st✐♠❛t❡ ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ L 2 (N ) t♦ L 2 ( N )✱ ❢♦r ❛•• f ∈ L 1 (N )✿ sup t>0 tλ t (T f ) ≤ c n,γ f 1 ❚♦ ♣r♦✈❡ t❤✐s t❤❡♦r❡♠✱ ✇❡ ✇✐•• ♥❡❡❞ ❛ ♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ r❡s✉•t ✇❤✐❝❤ ❝♦♥✲ st✐t✉t❡s t❤❡ ✜rst ♣❛rt ♦❢ t❤❡ ❝❤❛♣t❡r✳ ❚❤✐s ✐s ✇❤❡r❡ t❤❡ ♠♦st ✐♠♣♦rt❛♥t s✐♠✲ ♣•✐✜❝❛t✐♦♥s ❛r❡ ♠❛❞❡ ❝♦♠♣❛r❡❞ t♦ ❏✳ P❛r❝❡t✬s ✇♦r❦✳ ■♥ ♣❛rt✐❝✉•❛r✱ ♦✉r ♣r♦♦❢ ✐s ❡•❡♠❡♥t❛r② ❛♥❞ ❞♦❡s ♥♦t ❡①♣•✐❝✐t❡•② ✉s❡ t❤❡ s✐③❡ ❝♦♥❞✐t✐♦♥✳ ❲❡ ❞❡❝✐❞❡ t♦ ❞✐r❡❝t•② s❤♦✇ t❤❡ r❡s✉•t ✇✐t❤ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ✈❛r✐❛❜•❡s ❜✉t t❤❡ ♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ t❤❡♦✲ r❡♠ ✐s ❡ss❡♥t✐❛••② ❛ ❝♦♠♠✉t❛t✐✈❡ ♦♥❡✳ ❚❤❡ s❡❝♦♥❞ ♣❛rt✱ ✇❤✐❝❤ ✐s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠ ❤❛s ❜❡❡♥ s❤♦rt❡♥❡❞ ❛♥❞ ❝•❛r✐✜❡❞ t❤❛♥❦s t♦ ♠♦r❡ ❡✣❝✐❡♥t ♦r❣❛♥✐s❛✲ t✐♦♥ ❛♥❞ ❝♦♠♣✉t❛t✐♦♥s ❜✉t t❤❡ ✉♥❞❡r•②✐♥❣ ✐❞❡❛s ❛•• ❛♣♣❡❛r ✐♥ ❬✹✶(✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✇❡ ✉s❡ t❤❡ s❛♠❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✇❤✐❝❤ r❡•✐❡s ♦♥ ❈✉❝✉•❡s❝✉✬s ♣r♦•❡❝t✐♦♥s✳ ■t ✐s ❛ ♥❛t✉r❛• ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❝♦✉♥t❡r♣❛rt t♦ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✉s❡❞ ❢♦r t❤❡ ❝•❛ss✐❝❛• ♣r♦♦❢ ❜✉t ✉♥❢♦rt✉♥❛t❡•②✱ ♥❡✇ ❦✐♥❞ ♦❢ t❡r♠s ❛♣♣❡❛r ✇❤✐❝❤ ✇✐•• ♥❡❝❡ss✐t❛t❡ ♠♦r❡ r❡✜♥❡❞ ❡st✐♠❛t❡s ❛♥❞ ✐♥ ♣❛rt✐❝✉•❛r✱ t❤❡ ♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ t❤❡♦r❡♠✳ ■♥ ❛ t❤✐r❞ s❡❝t✐♦♥✱ ✇❡ s❤♦✇ ❛ ❜♦✉♥❞❡❞♥❡ss r❡s✉•t ❢♦r s✐♥❣✉•❛r ✐♥t❡❣r❛•s ✇✐t❤ ❍✐•❜❡rt ✈❛•✉❡❞ ❦❡r♥❡•s ❛♥❞ ♦♣❡r❛t♦r ✈❛•✉❡❞ ❢✉♥❝t✐♦♥s✳ ■t ✇❛s ❛•r❡❛❞② s❤♦✇♥ ✐♥ ❬✸✾( ❜② ❏✳ P❛r❝❡t ❛♥❞ ❚✳ ▼❡✐ ❜✉t ❢♦••♦✇s ❞✐r❡❝t•② ❢r♦♠ t❤❡♦r❡♠ ✹✳✶✳✺ ❛♥❞ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s✳ ❲❡ ❝♦♥❝•✉❞❡ t❤❡ ❝❤❛♣t❡r ❜② ♣r♦✈✐❞✐♥❣ ❛♥ L p ✲♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ t❤❡♦r❡♠ s✐♠✐✲ •❛r t♦ t❤❡ ♦♥❡ ♦❢ P✳ ❍②tö♥❡♥ ✐♥ ❬✷✷(✳ ❚❤❡ r❡s✉•t ❢♦••♦✇s ♠❛✐♥•② ❢r♦♠ ♦✉r ♣r♦♦❢ ♦❢ L 2 ✲♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥✱ ♠❛rt✐♥❣❛•❡ ✐♥❡q✉❛•✐t✐❡s ❛♥❞ ✐♥t❡r♣♦•❛t✐♦♥✳ ✽✻❈❍❆P❚❊| ✹✳ ❲❊❆❑ ❚(P❊ ■◆❊◗❯❆▲■❚( ❋❖| ❙■◆•❯▲❆| ■◆❚❊•|❆▲❙ ✹✳✶✳✷ ❆ t❡❝❤♥✐❝❛• r❡♠❛r❦ ❛♥❞ ♥♦t❛t✐♦♥s ❚❤❡ ❢♦••♦✇✐♥❣ r❡♠❛r❦ ❛••♦✇s ✉s t♦ ♠❛♥✐♣✉•❛t❡ t❤❡ ✐♥t❡❣r❛• ❡①♣r❡ss✐♦♥ ♦❢ T ✱ ✐♥ ♣❛rt✐❝✉•❛r t♦ ✉s❡ t❤❡ s♠♦♦t❤♥❡ss ❝♦♥❞✐t✐♦♥✳ |❡♠❛r❦ ✹✳✶✳✻✳ ❋♦r ❛♥② ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r T t❤❡r❡ ❡①✐sts ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r T ′ s✉❝❤ t❤❛t T f = T ′ f + gf ❢♦r ❛•• f ✐♥ L 2 (N ) ❛♥❞ T ′ ✐s t❤❡ str♦♥❣ •✐♠✐t ✐♥ B(L 2 (N )) ♦❢ ♦♣❡r❛t♦rs T i ❣✐✈❡♥ ❜②✿ T i f (x) = y∈R n k i (x, y)f (y)dy, ❢♦r ❛♥② f ∈ L 2 (N )✳ ❍❡r❡✱ k i ❝❛♥ ❜❡ t❛❦❡♥ ❛s ❛ tr✉♥❝❛t✐♦♥ ♦❢ k s♦ t❤❛t t❤❡ ✐♥t❡❣r❛• ♠❛❦❡s s❡♥s❡ ❛♥❞ g ✐s ❛ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥ ✐♥ N ✳ ❚❤✐s ❢❛❝t ✐s ❡①♣•❛✐♥❡❞ ✐♥ ♠♦r❡ ❞❡t❛✐•s ✐♥ ❬✶✼(✱ ♣r♦♣♦s✐t✐♦♥ 8.1.11✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ ✐t s✉✣❝❡s t♦ ♣r♦✈❡ t❤❡ t❤❡♦r❡♠ ❢♦r ♦♣❡r❛t♦rs ❣✐✈❡♥ ❜② ❛ ❝♦♥✈❡r❣✐♥❣ ✐♥t❡❣r❛• ❢♦r♠✉•❛ ♦♥ L 2 (N )✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠ ✇✐•• r❡•② ♦♥ t❤❡ ✉s❡ ♦❢ ❞②❛❞✐❝ ♠❛rt✐♥❣❛•❡s ✇❤✐❝❤ ✇✐•• r❡q✉✐r❡ ❛ ❢❡✇ ♥♦t❛t✐♦♥s✳ • ❙✐♥❝❡ ✇❡ ❞❡❛• ✇✐t❤ ❝✉❜❡s✱ t❤❡ ∞✲♥♦r♠ ✇✐•• ❜❡ ❡❛s✐❡r t♦ ♠❛♥✐♣✉•❛t❡ t❤❛♥ t❤❡ ❡✉❝•✐❞❡❛♥ ♦♥❡✳ ❍❡♥❝❡✱ ❢♦r ❛•• x ∈ R n ✱ ✇❡ s❡t |x| = x ∞ ✐♥ t❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤❡ t❡①t✳ • Q ✇✐•• ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❛•• ❞②❛❞✐❝ ❝✉❜❡s ❛♥❞ Q k t❤❡ s❡t ♦❢ ❞②❛❞✐❝ ❝✉❜❡s ♦❢ ❡❞❣❡ •❡♥❣t❤ 2 -k ✳ ▲❡t V k = 2 -nk ❜❡ t❤❡ ✈♦•✉♠❡ ♦❢ s✉❝❤ ❛ ❝✉❜❡✳ • ❋♦r ❛•• x ✐♥ R n ✱ Q x,k ✇✐•• ❞❡♥♦t❡ t❤❡ ❝✉❜❡ ✐♥ Q k ❝♦♥t❛✐♥✐♥❣ x ❛♥❞ c x,k ✐ts ❝❡♥t❡r✳ • ▲❡t (E k ) k∈Z ❜❡ t❤❡ ♠❛rt✐♥❣❛•❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❞②❛❞✐❝ ✜•tr❛t✐♦♥✱ ✐✳❡ ❢♦r ❛•• f ✿ E k (f )(x) = 1 V k Q x,k f (t)dt ❋♦r ❝♦♥✈❡♥✐❡♥❝❡✱ ✇❡ ✇✐•• ✇r✐t❡ f k := E k (f ) ❛♥❞ ∆ k (f ) := f k -f k-1 =: df k ✳ ❚❤❡ ✜•tr❛t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡s❡ ❡①♣❡❝t❛t✐♦♥s ✇✐•• ❜❡ ❞❡♥♦t❡❞ ❜② (N k ) k∈Z ✇❤❡r❡ N k ✐s t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ s✉❜❛•❣❡❜r❛ ♦❢ N ❝♦♥st✐t✉t❡❞ ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ❝♦♥st❛♥t ♦♥ ❝✉❜❡s ♦❢ ❡❞❣❡ •❡♥❣t❤ 2 -k ✳ • ❋♦r ❛♥② ♦❞❞ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r i ❛♥❞ Q ✐♥ Q k ✱ iQ ✇✐•• ❞❡s✐❣♥❛t❡ t❤❡ ✐♠❛❣❡ ♦❢ Q ❜② t❤❡ ❤♦♠♦t❤❡t② ♦❢ ❝❡♥t❡r c Q ❛♥❞ ♣❛r❛♠❡t❡r i s✉❝❤ t❤❛t iQ ✐s t❤❡ ✉♥✐♦♥ ♦❢ i n ❝✉❜❡s ✐♥ Q k ✳ • ◆♦t✐❝❡ t❤❛t ❢♦r ❛•• x, y ∈ R n ❛♥❞ k ∈ Z✱ x ∈ iQ y,k ⇔ y ∈ iQ x,k ✳ ❚❤❡ ♥♦t❛t✐♦♥ A B ✇✐•• st❛♥❞ ❢♦r ✧t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ n ❛♥❞ γ s✉❝❤ t❤❛t A ≤ cB✧✳ ■♥ t❤❡ ♥❡①t s❡❝t✐♦♥✱ ✇❡ ✇✐•• ❢r❡q✉❡♥t•② ✉s❡ ✧♣♦•❛r✧ ❝❤❛♥❣❡s ♦❢ ❝♦♦r❞✐♥❛t❡s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♥♦r♠ |.| s✐♥❝❡ ✐t ✐s ♠♦r❡ ❛❞❛♣t❡❞ t♦ ♦✉r ♣r♦❜•❡♠✳ ❚❤❡ s♣❤❡r✐❝❛• ❡•❡♠❡♥t ♦❢ ✈♦•✉♠❡ ✐s r❡♣•❛❝❡❞ ❜② t❤❡ ❜♦r❞❡r ♦❢ ❛ ❝✉❜❡ ✇❤✐❝❤ •❡❛❞s t♦ ❛ s✐♠✐•❛r ❢♦r♠✉•❛✿ R n f (|x|)dx = R + 2n(2r) n-1 f (r)dr✳ ✹✳✷✳ P❙❊❯❉❖✲▲❖❈❆▲■❙❆❚■❖◆ ✽✼ ✹✳✷ Ps❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ ✹✳✷✳✶ ❚❤❡♦r❡♠ ❆ •♦❝❛•✐s❛t✐♦♥ r❡s✉•t ✇♦✉•❞ ❜❡ ♦❢ t❤❡ ❢♦r♠ s✉♣♣ T f ≈ s✉♣♣ f ✇❤✐❝❤ ✐s ✐❞❡❛• t♦ s❤♦✇ ✇❡❛❦ ❜♦✉♥❞❡❞♥❡ss✳ ❚❤❡ t❤❡♦r❡♠ t❤❛t ❢♦••♦✇s ❡①♣r❡ss❡s ✐♥ ❛ ✇❛② t❤❡ ❢❛❝t t❤❛t s✐♥❣✉•❛r ✐♥t❡❣r❛•s r❛♣✐❞•② ✈❛♥✐s❤ ♦✉ts✐❞❡ ♦❢ t❤❡ s✉♣♣♦rt ♦❢ t❤❡ ❢✉♥❝t✐♦♥ t♦ ✇❤✐❝❤ t❤❡② ❛r❡ ❛♣♣•✐❡❞✳ ❆ s✐♠♣•❡r r❡s✉•t ♦❢ t❤✐s t②♣❡ ❛♣♣❡❛rs ✐♥ t❤❡ ❝♦♠✲ ♠✉t❛t✐✈❡ ♣r♦♦❢ ✐♥ t❤❡ L 1 ✲❝♦♥t❡①t ❛♥❞ ✐s ❡♥♦✉❣❤ t♦ ❝♦♥❝•✉❞❡ ✐♥ t❤✐s ❝❛s❡✳ ❇✉t✱ ❛s ♠❡♥t✐♦♥♥❡❞ ❜❡❢♦r❡✱ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❝❛s❡ r❡q✉✐r❡s ♥❡✇ t♦♦•s s✉❝❤ ❛s t❤❡ L 2 ✲✈❡rs✐♦♥ ♦❢ ♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ t❤❛t ❢♦••♦✇s✳ ◆♦t❡ t❤❛t t❤❡ ♣r♦♦❢ ✐s ✇r✐tt❡♥ ✇✐t❤ ♦♣❡r❛t♦r✲✈❛•✉❡❞ ❢✉♥❝t✐♦♥s ❜❡❝❛✉s❡ ✇❡ ✇✐•• ♥❡❡❞ t❤✐s r❡s✉•t •❛t❡r ❜✉t ✐s ❛•✲ ♠♦st ❛ ❝♦♣② ♦❢ t❤❡ ♣r♦♦❢ ✇❡ ❤❛❞ ✇✐t❤ s❝❛•❛r ❢✉♥❝t✐♦♥s✳ ❙♦✱ s✉♣r✐s✐♥❣•②✱ t❤❡ ♠♦st t❡❝❤♥✐❝❛• ♣❛rt ♦❢ t❤❡ ♣r♦♦❢ ♦❢ ♦✉r ♠❛✐♥ t❤❡♦r❡♠ ✐s ♣✉r❡•② ❝♦♠♠♠✉t❛t✐✈❡✳ ❚❤❡♦r❡♠ ✹✳✷✳✶✳ ▲❡t f ∈ L 2 (N ) ❛♥❞ s ∈ N✳ ❋ ♦r ❛•• k ∈ Z✱ •❡t A k ❛♥❞ B k ❜❡ ♣r♦•❡❝t✐♦♥s ✐♥ N k s✉❝❤ t❤❛t A ⊥ k df k+s = df k+s B ⊥ k = 0✳ ❋♦r ❛♥② ♦❞❞ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r d✱ ✇r✐t❡✿ A k = Q∈Q k A Q I Q ✱ A Q ∈ M ❛♥❞ ❞❡✜♥❡ dA k := Q∈Q k A Q I dQ ✳ ❉❡✜♥❡ dB k t❤❡ s❛♠❡ ✇❛②✳ ▲❡t✿ A f,s := k∈Z 5A k ❛♥❞ B f,s := k∈Z 5B k . ✭✹✳✶✮ ▲❡t T ❜❡ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❦❡r♥❡• k ✇✐t❤ ▲✐♣s❝❤✐t③ ♣❛r❛♠❡t❡r γ✱ ✈❡r✐❢②✐♥❣ t❤❡ s✐③❡ ❝♦♥❞✐t✐♦♥✱ t❛❦✐♥❣ ✈❛•✉❡s ✐♥ M ′ ∩ M ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ L 2 (N ) t♦ L 2 ( N )✳ ❚❤❡♥ ❢♦r ❛•• s ∈ N ❛♥❞ f ∈ L 2 (N ) ✇❡ ❤❛✈❡✿ A ⊥ f,s (T f ) 2 2 -γs 2 f 2 ❛♥❞ (T f )B ⊥ f,s 2 2 -γs 2 f 2 . ❚❤r♦✉❣❤♦✉t t❤❡ ❝♦✉rs❡ ♦❢ t❤❡ ♣r♦♦❢ ✇❡ ✇✐•• ♦❢t❡♥ ✉s❡ t❤❡ ❢♦••♦✇✐♥❣ •❡♠♠❛✿ ▲❡♠♠❛ ✹✳✷✳✷ ✭❙❝❤✉r✮✳ ▲❡t T ❜❡ ❛♥ ♦♣❡r❛t♦r ♦♥ L 2 (N ) ❣✐✈❡♥ ❜② ❛ M✲✈❛•✉❡❞ ❦❡r♥❡•✿ T f (x) = R n k(x, y)f (y)dy. ▲❡t S 1 (x) = R n k(x, y) M dy ❛♥❞ S 2 (y) = R n k(x, y) M dx t❤❡♥ ✿ T B(L2(N )) ≤ S 1 ∞ S 2 ∞ . Pr♦♦❢✳ ❚❤✐s ✐s ♥♦t ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❝♦♠♠✉t❛t✐✈❡ ❝❛s❡ ✭s❡❡ ❬✹✶(✮✳ ▲❡t f ∈ L 2 (N )✿ ✽✽❈❍❆P❚❊| ✹✳ ❲❊❆❑ ❚(P❊ ■◆❊◗❯❆▲■❚( ❋❖| ❙■◆•❯▲❆| ■◆❚❊•|❆▲❙ T f 2 2 = R n R n k(x, y)f (y)dy 2 2 dx ≤ R n R n k(x, y)f (y) 2 dy 2 dx ≤ R n R n k(x, y) M f (y) 2 dy 2 dx ≤ R n R n k(x, y) M dy R n k(x, y) M f (y) 2 2 dydx ≤ S 1 ∞ R n R n k(x, y) M f (y) 2 2 dydx ≤ S 1 ∞ S 2 ∞ R n f (y) 2 2 dy = S 1 ∞ S 2 ∞ f 2 2 ■t ✇✐•• ❛•s♦ ❜❡ ✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t ❜② ❝♦♥str✉❝t✐♦♥✱ ❢♦r ❛•• x, y ∈ R n s✉❝❤ t❤❛t x ∈ 5Q k,y ✱ ✇❡ ❤❛✈❡ A k (y) ≤ (5A k )(x) ≤ A f,s (x) ❛♥❞ A ⊥ k (y)df k+s (y) = 0✳ ❈♦♥s❡q✉❡♥t•②✿ A ⊥ f,s (x)df k+s (y) = (5A k ) ⊥ (x)df k+s (y) = 0. ✭✹✳✷✮ ❲❡ ✇✐•• ♦♥•② s❤♦✇ t❤❡ ♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ t❤❡♦r❡♠ ✐♥ t❤❡ ❝❛s❡ ♦❢ •❡❢t ♠✉•✲ t✐♣•✐❝❛t✐♦♥s s✐♥❝❡ t❤❡ ❡①❛❝t s❛♠❡ ♣r♦♦❢ ❝❛♥ ❜❡ ✇r✐t❡♥ ❢♦r r✐❣❤t ♠✉•t✐♣•✐❝❛t✐♦♥✳ ❆♥♦t❤❡r ✇❛② ♦❢ s❡❡✐♥❣ ✐t ✐s t❤❛t •❡❢t ❛♥❞ r✐❣❤t ♠✉•t✐♣•✐❝❛t✐♦♥ ❛r❡ ❡q✉✐✈❛•❡♥t ❜② t❛❦✐♥❣ ❛❞•♦✐♥ts✳ ❚❤❡ ❣❡♥❡r❛• str❛t❡❣② ✇✐•• ❜❡ t♦ ✜♥❞ ❛♥ ♦♣❡r❛t♦r T ′ ✈❡r✐❢②✐♥❣ A ⊥ f,s T f = A ⊥ f,s T ′ f ❛♥❞ s✉❝❤ t❤❛t ✇❡ ❝❛♥ ❝♦♥tr♦• T ′ B(L2(N )) t❤❛♥❦s t♦ t❤❡ •❡♠♠❛ ❛❜♦✈❡✳ ❇② r❡♠❛r❦ 4.1.6✱ ✇❡ ❝❛♥ ❛•s♦ s✉♣♣♦s❡ t❤❛t t❤❡ ✐♥t❡❣r❛• ❞❡✜♥✐♥❣ T ❝♦♥✈❡r❣❡s✳ ❚❤❡ r❡s✉•t ❢♦r ❛♥② T ❢♦••♦✇s t❤❡♥ ❞✐r❡❝t•② ❜② ❛♣♣r♦①✐♠❛t✐♦♥s✳ ✹✳✷✳✷ ❚❤❡ s s❤✐❢t ▲❡t f ∈ L 2 (N )✱ ✜①❡❞ t❤r♦✉❣❤♦✉t t❤❡ ♣r♦♦❢✳ ▲❡t s ∈ N✳ ❚♦ ♠❛❦❡ ✉s❡ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ A f,s ✇❡ ✐♠♠❡❞✐❛t❡•② ✇r✐t❡ T = k∈Z T ∆ k+s ✇❤❡r❡ t❤❡ s✉♠ ❝♦♥✈❡r❣❡s ❢♦r t❤❡ str♦♥❣ ♦♣❡r❛t♦r t♦♣♦•♦❣② ♦♥ B(L 2 (N ))✳ ❲❡ ✇✐•• s❤♦✇ ✐♥ t❤✐s s❡❝t✐♦♥ t❤❛t t❤❡ ❝♦♥st❛♥t 2 -γs ❛♣♣❡❛rs q✉✐t❡ ❡❛s✐•② t❤❛♥❦s t♦ t❤❡ s♠♦♦t❤♥❡ss ❝♦♥❞✐t✐♦♥ ✇❤❡♥ ❡st✐♠❛t✐♥❣ t❤❡ ♥♦r♠ ♦❢ T ∆ k+s f ✳ ❚❤❡ tr✐❝❦② ♣❛rt ✇✐•• ❜❡ t♦ ❣•✉❡ t❤❡s❡ ♣✐❡❝❡s ❜❛❝❦ t♦❣❡t❤❡r ✐♥ t❤❡ ❢♦••♦✇✐♥❣ s❡❝t✐♦♥s✳ ▲❡t k ∈ Z✱ ♥♦t❡ t❤❛t ✿ R n k(x, c y,k+s-1 )df k+s (y)dy = R n E k+s-1 k(x, c .,k+s-1 )df k+s (.) (y)dy = 0. ❚❤❡r❡❢♦r❡ ✇❡ ❝❛♥ ✇r✐t❡✿ A ⊥ f,s (x)T df k+s (x) = A ⊥ f,s (x) R n k(x, y)df k+s (y)dy = A ⊥ f,s (x) R n (k(x, y) -k(x, c y,k+s-1 ))df k+s (y)dy ✹✳✷✳ P❙❊❯❉❖✲▲❖❈❆▲■❙❆❚■❖◆ ✽✾ A ⊥ k (y) ❝♦♠♠✉t❡s ✇✐t❤ k(x, y) ❛♥❞ k(x, c y,k+s-1 ) s✐♥❝❡ k ✐s M ′ ✲✈❛•✉❡❞ s♦✿ A ⊥ f,s (x)T df k+s (x) = A ⊥ f,s (x) R n (k(x, y) -k(x, c y,k+s-1 ))(5A k ) ⊥ (x)df k+s (y)dy ❛♥❞ ❜② ✭✹✳✷✮✿ = A ⊥ f,s (x) R n I x / ∈5Q k,y (k(x, y) -k(x, c y,k+s-1 ))df k+s (y)dy. ❉❡♥♦t❡ ❜② T k t❤❡ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❦❡r♥❡• k k : (x, y) → I x / ∈5Q k,y k(x, y) ✭✹✳✸✮ ❛♥❞ T k,s t❤❡ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❦❡r♥❡• k k,s : (x, y) → I x / ∈5Q k,y (k(x, y) -k(x, c y,k+s-1 )). ✭✹✳✹✮ ■t ✇✐•• ❜❡ ✉s❡❢✉• t♦ ❡①♣r❡ss t❤❡ r❡s✉•t ♦❢ t❤❡ ♣r❡✈✐♦✉s ❝♦♠♣✉t❛t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡s❡ ♦♣❡r❛t♦rs✿ ▲❡♠♠❛ ✹✳✷✳✸✳ ❋♦r ❛•• k ∈ Z ❛♥❞ s ∈ N✿ A ⊥ f,s T ∆ k+s f = A ⊥ f,s T k ∆ k+s f = A ⊥ f,s T k,s ∆ k+s f, ✭✹✳✺✮ ❛♥❞ ♠♦r❡ ♣r❡❝✐s❡•②✿ T k ∆ k+s f = T k,s ∆ k+s f. ✭✹✳✻✮ ❚❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ T k,s ✐s ♠♦t✐✈❛t❡❞ ❜② t❤❡ ❢♦••♦✇✐♥❣ •❡♠♠❛✳ ▲❡♠♠❛ ✹✳✷✳✹✳ ❋♦r ❛•• k ∈ Z ❛♥❞ s ∈ N✿ T k,s B(L2(N )) 2 -sγ . ✭✹✳✼✮ Pr♦♦❢✳ ❚❤❡ s♠♦♦t❤♥❡ss ❤②♣♦t❤❡s✐s ♦♥ T ✭s❡❡ ❞❡✜♥✐t✐♦♥ ✹✳✶✳✸✮ ❣✐✈❡s✿ k k,s (x, y) M = I x / ∈5Q k,y k(x, y) -k(x, c y,k+s-1 ) M ≤ I x / ∈5Q k,y |y -c y,k+s-1 | γ |y -x| n+γ I x / ∈5Q k,y 2 -(k+s)γ |y -x| n+γ . ✭✹✳✽✮ ❚❤❡ ❝♦♥❞✐t✐♦♥ |y -c y,k+s-1 | ≤ 1 2 |x -y| ✐s ✈❡r✐✜❡❞ ❡✈❡♥ ❢♦r s = 0 ❛s •♦♥❣ ❛s x / ∈ 5Q k,y ✳ ❚❤✐s ❡st✐♠❛t❡ ✐s ❡♥♦✉❣❤ t♦ ❛♣♣•② ❙❝❤✉r✬s •❡♠♠❛ ❛♥❞ ✇❡ ♦❜t❛✐♥ t❤❡ r❡s✉•t ❜② ❛ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ✉s✐♥❣ ❛ ✧♣♦•❛r✧ ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s✳ ✾✵❈❍❆P❚❊| ✹✳ ❲❊❆❑ ❚(P❊ ■◆❊◗❯❆▲■❚( ❋❖| ❙■◆•❯▲❆| ■◆❚❊•|❆▲❙ ✹✳✷✳✸ ❖♥❡ ♠♦r❡ ❝❛♥❝❡••❛t✐♦♥ ♣r♦♣❡rt② ❲❡ ✐♥tr♦❞✉❝❡ ❛♥♦t❤❡r ❦❡r♥❡• ♠♦❞✐✜❝❛t✐♦♥ ✇❤♦s❡ ♣✉r♣♦s❡ ✐s ♥♦t ✐♠♠❡❞✐❛t❡•② ❝•❡❛r ❜✉t ✇✐•• ❜❡ ❝r✉❝✐❛• ✐♥ ♦❜t❛✐♥✐♥❣ t❤❡ ❡st✐♠❛t❡s t♦ ❛♣♣•② ❙❝❤✉r✬s •❡♠♠❛ •❛t❡r✳ Pr♦♣♦s✐t✐♦♥ ✹✳✷✳✺✳ ❋♦r ❛•• k ∈ Z ❛♥❞ s ∈ N t❤❡r❡ ❡①✐sts ❛♥ ♦♣❡r❛t♦r S k,s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❦❡r♥❡• s k,s s✉❝❤ t❤❛t t❤❡ t❤r❡❡ ❢♦••♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❤♦•❞✿ ✶✳ A ⊥ f,s T ∆ k+s f = A ⊥ f,s S k,s ∆ k+s f ✱ ✷✳ y∈R n s k,s (x, y)dy = 0 ❢♦r ❛•• x ∈ R n ✱ ✸✳ k,s (x, y) M I |x-y|>2 -k-1 2 -(k+s)γ |y -x| n+γ ✳ Pr♦♦❢✳ ❈♦♥s✐❞❡r R k,s ❛♥ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❦❡r♥❡• r k,s ❞❡✜♥❡❞ ❜②✿ r k,s (x, y) := I A k K(x)2 -(k+s)γ I n,γ |y -x| n+γ ✇❤❡r❡ A k := {(x, y) : x ∈ 5Q y,k , x / ∈ 3Q y,k }✱ K(x) = -R n k k,s (x, t)dt ❛♥❞ I n,γ = R n I A k (0, t) 2 -(k+s)γ |t| n+γ dt✳ ❲❡ ❝•❛✐♠ t❤❛t S k,s := T k,s + R k,s s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s ❛❜♦✈❡✳ ❈♦♥❞✐t✐♦♥ 1✳ A ⊥ f,s R k,s ∆ k+s f = 0 ✐s ✈❡r✐✜❡❞ s✐♥❝❡✿ s✉♣♣ r k,s ⊂ {(x, y) : x ∈ 5Q y,k } ✐✳❡ r k,s = I x∈5Q y,k r k,s ✳ ■♥❞❡❡❞✱ t❤❡ s❤✐❢t s❡❝t✐♦♥ s❤♦✇s t❤❛t t♦ ❝♦♠♣✉t❡ A ⊥ f,s R k,s ∆ k+s f ✱ ❛♥❞ ✐♥ ♣❛rt✐❝✉✲ •❛r (5A k ) ⊥ R k,s ∆ k+s f ✱ ✇❡ ♠✐❣❤t ❛s ✇❡•• r❡♣•❛❝❡ t❤❡ ❦❡r♥❡• r k,s ❜② I x / ∈5Q k,y r k,s = I x / ∈5Q k,y I x∈5Q y,k r k,s = 0✳ ❈♦♥❞✐t✐♦♥ 2✳ ■t ✐s ❞✐r❡❝t ❜② ❞❡✜♥✐t✐♦♥ ♦❢ K ❛♥❞ I n,γ ✳ ▲❡t x ∈ R n ✿ y∈R n s k,s (x, y)dy = R n k k,s (x, y)dy + K(x) = 0✳ ❈♦♥❞✐t✐♦♥ 3✳ ■t s✉✣❝❡s t♦ s❤♦✇ t❤❛t K(x) M ✐s ❜♦✉♥❞❡❞ ❜② ❛ ❝♦♥st❛♥t ❞❡♣❡♥❞✐♥❣ ♦♥•② ♦♥ n ❛♥❞ γ✳ ❇② ❛ ✧♣♦•❛r✧ ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c n s✉❝❤ t❤❛t✿ K(x) M ≤ |t|>4.2 -k 2 -(k+s)γ |t| n+γ dt = c n 2 -γs ✳ ❚❤❡ ❜♦✉♥❞ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ x✱ ❛s ✇❡ ♥❡❡❞❡❞✳ ✹✳✷✳ P❙❊❯❉❖✲▲❖❈❆▲■❙❆❚■❖◆ ✾✶ ✹✳✷✳✹ ❚❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❲❡ ❛r❡ r❡❛❞② t♦ ❞❡❝♦♠♣♦s❡ T ✐♥t♦ ♦♣❡r❛t♦rs ✇❤♦s❡ ♥♦r♠s ✐♥ B(L 2 (N )) ❛r❡ ❝♦♥tr♦••❡❞✳ ❋✐① s ∈ N✱ ✇r✐t❡✿ A ⊥ f,s T f = A ⊥ f,s k∈Z T ∆ k+s f = A ⊥ f,s k∈Z (T k + R k,s )∆ k+s f = A ⊥ f,s k∈Z i∈Z ∆ k+i (T k + R k,s )∆ k+s f ◆♦t❡ t❤❛t ❢♦r i ≥ 1✱ (5A k ) ⊥ ❝♦♠♠✉t❡s ✇✐t❤ ∆ k+i ❛♥❞ r❡❝❛•• t❤❛t t❤❡ ✜rst ♣♦✐♥t ♦❢ R k,s ✬s ❝♦♥str✉❝t✐♦♥ ✐♠♣•✐❡s t❤❛t (5A k ) ⊥ R k,s ∆ k+s f = 0✱ t❤❡♥✿ A ⊥ f,s ∆ k+i (T k + R k,s )∆ k+s f = A ⊥ f,s ∆ k+i (5A k ) ⊥ (T k + R k,s )∆ k+s f = A ⊥ f,s ∆ k+i T k ∆ k+s f ❋♦r ❛♥② i ∈ Z✱ ✇❡ ❤❛✈❡✿ A ⊥ f,s ∆ k+i (T k + R k,s )∆ k+s f = A ⊥ f,s ∆ k+i (T k,s + R k,s )∆ k+s f = A ⊥ f,s ∆ k+i S k,s ∆ k+s f ◆♦✇ ✇❡ ❝❛♥ ✇r✐t❡ ♦✉r ✜♥❛• ❞❡❝♦♠♣♦s✐t✐♦♥✿ A ⊥ f,s T f = A ⊥ f,s ∞ i=0 Φ i f + ∞ i=1 Ψ i f ✭✹✳✾✮ ✇❤❡r❡✿ Φ i = k∈Z ∆ k+i T k ∆ k+s ❛♥❞ Ψ i = k∈Z ∆ k-i S k,s ∆ k+s ❲❡ ❤❛✈❡ t❤❡♥ r❡❞✉❝❡❞ ♦✉r ♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ t❤❡♦r❡♠ t♦ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✳ Pr♦♣♦s✐t✐♦♥ ✹✳✷✳✻✳ ❚❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡s ❤♦•❞ ❢♦r i ≥ 0✿ Φ i B(L2(N )) 2 -γ i+s 2 ❛♥❞ Ψ i B(L2(N )) √ 1 + i2 -γ i+s 2 ❚❤❡ ♣r♦♦❢ ♦❢ t❤✐s r❡s✉•t ✇✐•• ♦❝❝✉♣② t❤❡ ♥❡①t t✇♦ ♣❛rts✳ ✹✳✷✳✺ ❊st✐♠❛t❡ ❢♦r Φ i ▲❡t i ≥ 0✳ ◆♦t❡ t❤❛t T * ✐s ❛•s♦ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r✱ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❦❡r♥❡• k ⋆ ✇❤❡r❡✿ k ⋆ (x, y) = k(y, x) * ✳ ❙♦ k ⋆ s❛t✐s✜❡s t❤❡ s♠♦♦t❤♥❡ss ❝♦♥❞✐t✐♦♥ ♦❢ ♣❛r❛♠❡t❡r γ✱ ✇❤✐❝❤ ♠❡❛♥s ❜② (4.7) t❤❛t✿ (T * ) k,s B(L2(N )) 2 -sγ ✳ ✾✷❈❍❆P❚❊| ✹✳ ❲❊❆❑ ❚(P❊ ■◆❊◗❯❆▲■❚( ❋❖| ❙■◆•❯▲❆| ■◆❚❊•|❆▲❙ ❈♦♥s❡q✉❡♥t•② ✿ ∆ k+i T k B(L2(N )) = T * k ∆ k+i B(L2(N )) = (T * ) k,i ∆ k+i B(L2(N )) 2 -γi ✱ ✇❤❡r❡ ✇❡ ✉s❡❞ t❤❛t (T k ) * = (T * ) k ❛♥❞ (4.6)✳ ❇② ♦rt❤♦❣♦♥❛•✐t② ♦❢ t❤❡ ∆ k ✱ ✇❡ ❤❛✈❡ ♦♥ ♦♥❡ ❤❛♥❞✿ Φ i B(L2(N )) ≤ sup k ∆ k+i T k ∆ k+s B(L2(N )) ≤ sup k ∆ k+i T k B(L2(N )) 2 -γi ✱ ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ ✿ Φ i B(L2(N )) ≤ sup k ∆ k+i T k ∆ k+s B(L2(N )) ≤ sup k T k ∆ k+s B(L2(N )) 2 -γs ✳ ❇② ❝♦♠❜✐♥✐♥❣ t❤❡ t✇♦✱ Φ i B(L2(N )) 2 -γ s+i 2 ✳ ✹✳✷✳✻ ❊st✐♠❛t❡ ❢♦r Ψ i ▲❡♠♠❛ ✹✳✷✳✼✳ ❋♦r ❛•• x ∈ R n ✱ k ∈ Z ❛♥❞ i ∈ N✱ t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡ ❤♦•❞s✿ t∈Q x,k-i y∈Q c x,k-i I |t-y|>2 1-k 1 |t -y| n+γ dydt (1 + i)2 (γ-n)(k-i) ✳ Pr♦♦❢✳ ❋✐① x ∈ R n ✱ k ∈ Z ❛♥❞ i ∈ N✳ ❉❡♥♦t❡ Q = Q x,k-i ✱ c t❤❡ ❝❡♥t❡r ♦❢ Q ❛♥❞✿ X = t∈Q y∈Q c I |t-y|>2 1-k 1 |t -y| n+γ dydt✳ ❋♦r ❡✈❡r② t ✐♥ Q •❡t δ t ❜❡ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t t♦ Q c ❛♥❞ ♥♦t✐❝❡ t❤❛t ❢♦r ❛♥② t✿ Q c I |t-y|>2 1-k 1 |t -y| n+γ dy ≤ |y-t|>δt I |t-y|>2 1-k 1 |t -y| n+γ dy =: f (δ t ) ■♥❞❡❡❞✱ t❤❡ t❡r♠ ♦♥ t❤❡ r✐❣❤t ♦♥•② ❞❡♣❡♥❞s ♦♥ δ t ✳ ❋♦r r ≤ 2 -(k-i+1) ✿ {t ∈ Q : δ t = r} = {t ∈ Q : |t -c| = 2 -(k-i+1) -r =: r ′ }✱ ❙♦ ❜② ❛ ✧♣♦•❛r✧ ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s✿ X ≤ 2 -(k-i+1) 0 2n(2r ′ ) n-1 f (r)dr 2 -(k-i)(n-1) 2 -(k-i+1) 0 f (r)dr✳ ❇② ❛ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✿ f (r) r -γ ❢♦r ❛•• r 2 γ(k-1) ✐❢ ♠♦r❡♦✈❡r r ≤ 2 -(k-1) ❋♦r 0 < γ < 1 ✇❡ ♦♥•② ♥❡❡❞ t❤❡ ✜rst ❡st✐♠❛t❡✿ 1 r dr 1 + i ❚❤❡r❡❢♦r❡✿ X 2 -(k-i)(n-1) (1 + i) = (1 + i)2 -n(k-i) 2 γ(k-i) ✳ Pr♦♣♦s✐t✐♦♥ ✹✳✷✳✽✳ ❋♦r ❛•• k ✐♥ Z ❛♥❞ i > 0✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡✿ E k-i S k,s B(L2(N )) √ 1 + i2 -γ(s+i/2) Pr♦♦❢✳ ❋♦r ❛♥② ❢✉♥❝t✐♦♥ g : E k-i S k,s g(x) = Q x,k-i 1 V k-i R n s k,s (t, y)g(y)dydt✳ ❙♦ E k-i S k,s ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❦❡r♥❡•✿ E : (x, y) → 1 V k-i Q x,k-i s k,s (t, y)dt = 1 V k-i Q c x,k-i s k,s (t, y)dt✱ ✇❤❡r❡ ✇❡ ✉s❡❞ t❤❡ ❝❛♥❝❡••❛t✐♦♥ ♣r♦♣❡rt② ♦♥ S k,s ✳ ❲❡ ❛r❡ •♦♦❦✐♥❣ t♦ ❡st✐♠❛t❡ ❜♦t❤ ✐♥t❡❣r❛•s ✐♥ ♦r❞❡r t♦ ❛♣♣•② ❙❝❤✉r✬s •❡♠♠♠❛✳ ❋✐① x✿ R n E(x, y) M dy = Q x,k-i E(x, y) M dy + Q c x,k-i E(x, y) M dy = Q x,k-i Q c x,k-i 1 V k-i s k,s (t, y)dt M dy + Q c x,k-i Q x,k-i 1 V k-i s k,s (t, y)dt M dy 1 V k-i Q x,k-i Q c x,k-i I |t-y|>2 1-k 2 -γ(k+s) |t -y| n+γ dtdy ❯s✐♥❣ •❡♠♠❛ 4.2.7 ✿ R n E(x, y) M dy 1 V k-i (1 + i)2 -n(k-i) 2 γ(k-i) 2 -γ(k+s) (1 + i)2 -γ(i+s) ✾✹❈❍❆P❚❊| ✹✳ ❲❊❆❑ ❚(P❊ ■◆❊◗❯❆▲■❚( ❋❖| ❙■◆•❯▲❆| ■◆❚❊•|❆▲❙ ❚❤❡ ♦t❤❡r ❡st✐♠❛t❡✱ ✇✐t❤ y ✜①❡❞✱ ✐s str❛✐❣❤t❢♦r✇❛r❞✿ R n E(x, y) M dx = R n 1 V k-i Q x,k-i s k,s (t, y)dt M dx ≤ R n 1 V k-i Q x,k-i s k,s (t, y) M dtdx = R n s k,s (t, y) M dt 2 -γs ❚❤❡ ♣r♦♣♦s✐t✐♦♥ ❢♦••♦✇s ❞✐r❡❝t•② ❢r♦♠ t❤❡ t✇♦ ♣r❡✈✐♦✉s ❝♦♠♣✉t❛t✐♦♥s ❛♥❞ ❙❝❤✉r✬s •❡♠♠❛✳ ❲❡ ♥♦✇ ❤❛✈❡✿ ∆ k-i S k,s ∆ k+s B(L2(N )) ≤ E k-i S k,s B(L2(N )) √ 1 + i 2 -γ s+i 2 ✳ ❲❡ ❝♦♥❝•✉❞❡ ♦♥❝❡ ♠♦r❡ t❤❛♥❦s t♦ t❤❡ ♦rt❤♦❣♦♥❛•✐t② ❜❡t✇❡❡♥ t❤❡ ∆ k-i S k,s ∆ k+s ✿ Ψ i B(L2(N )) sup k ∆ k-i S k,s ∆ k+s B(L2(N )) √ 1 + i 2 -γ s+i 2 ❚❤✐s ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ t❤❡♦r❡♠✳ ✹✳✸ Pr♦♦❢ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠ ✹✳✸✳✶ ❚❤❡ ❣♦♦❞ ❛♥❞ t❤❡ ❜❛❞ ❢✉♥❝t✐♦♥s ❚❤❡ ✐❞❡❛ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ❞❡❝♦♠♣♦s✐t✐♦♥ ❝♦♠❡s ❢r♦♠ t❤❡ ❝•❛ss✐❝❛• ♣r♦♦❢ ♦❢ t❤❡ ✇❡❛❦ t②♣❡ ✐♥❡q✉❛•✐t② ❢♦r s✐♥❣✉•❛r ✐♥t❡❣r❛•s✳ ■t ❤❛s ❜❡❡♥ ♥♦t✐❝❡❞ t❤❛t t❤❡ ❝♦♠✲ ♠✉t❛t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ♠❛rt✐♥❣❛•❡s ✇❤✐❝❤ ✐s ✇❡•• s✉✐t❡❞ ❢♦r ❛ tr❛♥s•❛t✐♦♥ ✐♥ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s❡tt✐♥❣✳ ❍♦✇❡✈❡r✱ ❡✈❡♥ ✇✐t❤ t❤✐s ✐❞❡❛✱ t❤❡ ❝♦♥str✉❝t✐♦♥ ✐s ♥♦t ✐♠♠❡❞✐❛t❡ ❛♥❞ t❤❡ ❡st✐♠❛t❡s ❛r❡ ♠♦r❡ ❞✐✣❝✉•t t♦ ♦❜t❛✐♥ ❞✉❡ t♦ t❤❡ ❛♣♣❡❛r❛♥❝❡ ♦❢ ♥❡✇ ✧♦✛✲❞✐❛❣♦♥❛•✧ t❡r♠s✳ ❋✐① t > 0 ❛♥❞ f ∈ L 1 (N )✳ ❲❡ ✇✐•• s✉♣♣♦s❡ t❤❛t f ✐s ♣♦s✐t✐✈❡ t♦ ❛✈♦✐❞ ✉♥♥❡❝❡ss❛r② ❝♦♠♣✉t❛t✐♦♥s✳ ❚❤✐s ✐s ♣♦ss✐❜•❡ ❜❡❝❛✉s❡ f ❝❛♥ ❛•✇❛②s ❜❡ ✇r✐tt❡♥ ❛s✿ f = f 1 -f 2 + if 3 -if 4 ✇✐t❤ f k 1 ≤ f 1 ❢♦r k = 1, 2, 3, 4✳ ❙✉♣♣♦s❡ ❛•s♦ t❤❛t {x ∈ R n : f (x) = 0} ✐s ❜♦✉♥❞❡❞ ✭✐♥ ♦t❤❡r ✇♦r❞s✱ f ✱ ❝♦♥s✐❞❡r❡❞ ❛s ❛ ❢✉♥❝t✐♦♥✱ ❤❛s ❛ ❝♦♠♣❛❝t s✉♣♣♦rt✮✱ ❛♥❞ t❤❛t f ✐s ✐♥ L 2 (N ) ∩ N ✳ ❲❡ ❝❛♥ ♠❛❦❡ t❤❡s❡ ❛ss✉♠♣t✐♦♥s s✐♥❝❡ t❤❡ ❢✉♥❝t✐♦♥s s❛t✐s❢②✐♥❣ t❤❡♠ ❢♦r♠ ❛ ❞❡♥s❡ s✉❜s♣❛❝❡ S ♦❢ L 1 (N )✳ ❖♥❝❡ t❤❡ t❤❡♦r❡♠ ✐s ♣r♦✈❡♥✱ T ❝❛♥ ❜❡ ❞❡✜♥❡❞ ♦♥ L 1 (N ) ❛s t❤❡ ♦♥•② ❜♦✉♥❞❡❞ ❡①t❡♥s✐♦♥ ♦❢ ✐ts r❡str✐❝t✐♦♥ t♦ S✳ ❉❡♥♦t❡ ❜② (f n ) n∈Z t❤❡ ♠❛rt✐♥❣❛•❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ f ❛♥❞ t❤❡ ✜•tr❛t✐♦♥ (N n ) n∈Z ✳ ❚❤❡ ♠❛✐♥ t♦♦•s t♦ ❞❡❝♦♠♣♦s❡ f ✐♥t♦ ❛ ❣♦♦❞ ❛♥❞ ❛ ❜❛❞ ♣❛rt ❛r❡ ❈✉❝✉•❡s❝✉✬s ♣r♦✲ •❡❝t✐♦♥s✿ ❚❤❡♦r❡♠ ✹✳✸✳✶ ✭❈✉❝✉•❡s❝✉✮✳ ▲❡t x = (x n ) ❜❡ ❛ ❜♦✉♥❞❡❞ ♣♦s✐t✐✈❡ L 1 ✲♠❛rt✐♥❣❛•❡ ❛♥❞ t ≥ 0✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ (q n ) ♦❢ ♣r♦•❡❝t✐♦♥s ✐♥ N s✉❝❤ t❤❛t ❢♦r ❡✈❡r② n ≥ 1✿ ✶✳ q n ∈ N n ✳ ✷✳ q n ❝♦♠♠✉t❡s ✇✐t❤ q n-1 x n q n-1 ✳ ✸✳ q n x n q n ≤ t✳ ✹✳ ♠♦r❡♦✈❡r✱ ✐❢ q = q n t❤❡♥ ✿ qx n q ≤ t ❢♦r n ≥ 1 ❛♥❞ τ (q ⊥ ) ≤ x 1 t ✳ ❚❤❡ ❜♦✉♥❞❡❞♥❡ss ❤②♣♦t❤❡s✐s ♦♥ f ❛♥❞ ✐ts s✉♣♣♦rt ✐♠♣•② t❤❛t t❤❡r❡ ❡①✐sts n 0 s✉❝❤ t❤❛t ❢♦r ❛•• n ≤ n 0 ✱ f n ≤ t✳ ❲✐t❤♦✉t •♦ss ♦❢ ❣❡♥❡r❛•✐t②✱ ✇❡ ✇✐•• s✉♣♣♦s❡ t❤❛t n 0 = 0✳ ❋r♦♠ ♥♦✇ ♦♥✱ •❡t q n ❞❡♥♦t❡ t❤❡ ♣r♦•❡❝t✐♦♥s ❣✐✈❡♥ ❜② ❈✉❝✉•❡s❝✉✬s t❤❡♦r❡♠✱ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t ❛♥❞ (f n ) n≥0 ✳ ❚♦ ❝♦♠♣•❡t❡ t❤✐s ❞❡✜♥✐t✐♦♥ •❡t q n = 1 ❢♦r ❛•• n ≤ 0✳ ◆♦t✐❝❡ t❤❛t ❢♦r ❛•• n ∈ Z✱ q n f n q n ≤ t✳ ❉❡✜♥❡ ✿ ∀n ∈ Z, p n = q n-1 -q n ❛♥❞ p ∞ = q✳ ▲❡t Z = Z ∪ {∞}✳ ❇② ❞❡✜♥✐t✐♦♥✱ p n ∈ N n ❛♥❞✿ n∈Z p n = 1✳ ❲❤✐❝❤ ❛••♦✇s ✉s t♦ ❞❡✜♥❡ t❤❡ ❣♦♦❞ ❛♥❞ ❜❛❞ ♣❛rts ❛s ❢♦••♦✇s ✿ g = i∈Z j∈Z p i f i∨j p j ❛♥❞ b = i∈Z j∈Z p i (f -f i∨j )p j ✳ ❇② ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✭s❡❡ ❬✶✻(✮✿ λ t (T f ) λ t/2 (T g) + λ t/2 (T b) ❙♦ ✐t s✉✣❝❡s t♦ ♣r♦✈❡ ❡st✐♠❛t❡s ♦❢ t❤❡ ❢♦r♠✿ λ t (T x) f 1 t ❢♦r ❜♦t❤ x = g ❛♥❞ x = b✳ ❚❤✐s ✐s t❤❡ ♣✉r♣♦s❡ ♦❢ t❤❡ ♥❡①t s❡❝t✐♦♥s✳ |❡♠❛r❦ ✹✳✸✳✷✳ ❚❤❡ s❛♠❡ ❢♦r♠✉•❛ ❢♦r b ❛♥❞ g ✇♦r❦s ✐♥ t❤❡ ❝♦♠♠✉t❛t✐✈❡ ❝❛s❡ ❡①❝❡♣t t❤❛t ♦♥•② t❤❡ ❞✐❛❣♦♥❛• t❡r♠s ❛r❡ ♥♦♥ ③❡r♦✳ ❚❤✐s ❡①♣•❛✐♥s ✇❤② t❤❡ ❝♦♠♠✉✲ t❛t✐✈❡ ♣r♦♦❢ ❝❛♥ ♦♥•② ❜❡ r❡♣❡❛t❡❞ ❢♦r t❤❡ ❞✐❛❣♦♥❛• t❡r♠s ❛♥❞ ✧s❤✐❢t❡❞✧ ❞✐❛❣♦♥❛•s✳ ❲❤❛t ♠❛❦❡s t❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥ ✇♦r❦ ✐s t❤❛t✱ ❞✉❡ t♦ t❤❡ ♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ •❡♠♠❛✱ t❤❡ ❡st✐♠❛t❡s ❢♦r ✧s❤✐❢t❡❞✧ ❞✐❛❣♦♥❛•s ❣❡t ❡①♣♦♥❡♥t✐❛••② ❜❡tt❡r ❛s t❤❡ s❤✐❢t ✐♥❝r❡❛s❡s✳ ❲❡ ✇✐•• ✉s❡ t❤❡ ❢♦••♦✇✐♥❣ ♥♦t❛t✐♦♥✿ ❢♦r ❛•• k ❛♥❞ Q ∈ Q k ✱ p Q := p k (x) ❢♦r ❛♥② x ∈ Q✳ ❲❡ ✇✐•• ❛•s♦ ♥❡❡❞ t✇♦ •❡♠♠❛s ✇❤✐❝❤ ❛r❡ ❞✐r❡❝t•② ❞❡❞✉❝❡❞ ❢r♦♠ t❤❡ ❝♦♥str✉❝t✐♦♥✳ ▲❡♠♠❛ ✹✳✸✳✸✳ ❋♦r ❛•• k ∈ Z✱ ✇❡ ❤❛✈❡ ✿ f k+1 ≤ 2 n f k ✳ Pr♦♦❢✳ ❚❤✐s ✐s str❛✐❣❤t❢♦r✇❛r❞ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ f k ❛♥❞ ♣♦s✐t✐✈✐t② ♦❢ f ✳ ▲❡t x ∈ R n ✿ 2 n f k (x) = 2 n V k Q x,k f (t)dt ≥ 2 n V k Q x,k+1 f (t)dt = f k+1 (x)✳ ▲❡♠♠❛ ✹✳✸✳✹✳ ▲❡t d ❜❡ ❛♥ ♦❞❞ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❉❡✜♥❡✿ ζ d = Q∈Q p Q I dQ ⊥ ✳ ❚❤❡♥ ✿ ✶✳ τ (ζ ⊥ d ) ≤ d n f 1 t ✷✳ ❋♦r ❛•• ❝✉❜❡s Q ∈ Q✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❝❛♥❝❡••❛t✐♦♥ ♣r♦♣❡rt②✿ x ∈ dQ → ζ d (x)p Q = p Q ζ d (x) = 0✳ Pr♦♦❢✳ ❚❤❡ ✜rst ❡st✐♠❛t❡ ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❈✉❝✉•❡s❝✉✬s ✐♥❡q✉❛•✐t②✿ τ (ζ ⊥ d ) ≤ Q∈Q d n τ (p Q I Q ) = d n ∞ k=1 τ (p k ) = d n τ (q ⊥ ) ≤ d n f 1 t ✳ ▲❡t Q ∈ Q ❛♥❞ x ∈ dQ✳ ❇② ❝♦♥str✉❝t✐♦♥✱ ζ ⊥ d (x) ≥ p Q s♦ ζ d (x) ≤ p ⊥ Q ✳ ❚❤✐s ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢✳ ❋r♦♠ ♥♦✇ ♦♥✱ ✇❡ ✜① d = 5 ❛♥❞ ❞❡♥♦t❡ ζ d ❜② ζ✳ |❡♠❛r❦ ✹✳✸✳✺✳ ❚❤✐s ♣r♦•❡❝t✐♦♥ ζ ✐s t♦ ❜❡ t❤♦✉❣❤t ❛s ❛ ❞✐•❛t❛t✐♦♥ ♦❢ t❤❡ s✉♣♣♦rt ♦❢ t❤❡ ❜❛❞ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ❛•r❡❛❞② ♣•❛②s ❛ ❝r✉❝✐❛• r♦•❡ ✐♥ t❤❡ ❝♦♠♠✉t❛t✐✈❡ s❡tt✐♥❣✳ ✹✳✸✳✷ ❊st✐♠❛t❡ ❢♦r t❤❡ ❜❛❞ ❢✉♥❝t✐♦♥ ❚❤❡ str❛t❡❣② ♦❢ ♣r♦♦❢ ✐s t♦ ✇r✐t❡✿ T b = ζT bζ + (1 -ζ)T bζ + ζT b(1 -ζ) + (1 -ζ)T b(1 -ζ)✳ ❚❤❡r❡❢♦r❡✱ •❡♠♠❛ 4.3.4 ❛♥❞ ❚❝❤❡❜②❝❤❡✈✬s ✐♥❡q✉❛•✐t② ❣✐✈❡✿ λ t (T b) τ (1 -ζ) + λ t (ζT bζ) f 1 t + ζT bζ 1 t ✳ ❚❤❡ ❡st✐♠❛t❡ ❢♦r t❤❡ ❜❛❞ ❢✉♥❝t✐♦♥ ✐s ♥♦✇ r❡❞✉❝❡❞ t♦ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐✲ t✐♦♥✿ Pr♦♣♦s✐t✐♦♥ ✹✳✸✳✻✳ ❲❡ ❤❛✈❡ t❤❡ ❡st✐♠❛t❡✿ ζT bζ 1 f 1 ✳ ✹✳✸✳ P|❖❖❋ ❖❋ ❚❍❊ ▼❆■◆ ❚❍❊❖|❊▼ ✾✼ ❚❤❡ ♣r♦♦❢ ✇✐•• r❡q✉✐r❡ t❤r❡❡ ✐♥t❡r♠❡❞✐❛t❡ •❡♠♠❛s✳ ❉❡✜♥❡✱ ❢♦r ❛•• i, j ∈ Z ✿ b i,j = p i (f -f i∨j )p j ✳ ▲❡♠♠❛ ✹✳✸✳✼✳ ❋♦r ❛•• s ∈ Z ✿ i-j=s b i,j 1 f 1 Pr♦♦❢✳ ▲❡t i, j ∈ Z ✿ b i,j 1 ≤ p i f p j 1 + p i f i∨j p j 1 ❇② ❍♦•❞❡r✬s ✐♥❡q✉❛•✐t② ✿ b i,j 1 ≤ f 1/2 p i 2 f 1/2 p j 2 + f 1/2 i∨j p i 2 f 1/2 i∨j p j 2 ≤ 1 2 p i f p i 1 + p j f p j 1 + p i f i∨j p i 1 + p j f i∨j p j 1 ≤ τ (p i f ) + τ (p j f ) ❈♦♥s❡q✉❡♥t•②✱ ❢♦r ❛•• s ∈ Z✿ i-j=s b i,j 1 ≤ 2 i∈Z τ (p i f ) f 1 ▲❡♠♠❛ ✹✳✸✳✽✳ ❚❤❡ ❢♦••♦✇✐♥❣ ❝❛♥❝❡••❛t✐♦♥ ♣r♦♣❡rt✐❡s ❤♦•❞✿ • ❢♦r ❛•• i, j ∈ Z ❛♥❞ Q ∈ Q i∨j ✿ Q b i,j = 0❀ • ❢♦r ❛•• x,
g (d) = i∈Z p i f i p i ✱ g (l)
= i<j∈Z p i f j p j + q ⊥ f q ❛♥❞ Pr♦♦❢ ♦❢ t❤❡ g (d) ❡st✐♠❛t❡✳ ❙✐♥❝❡ T ✐s ❜♦✉♥❞❡❞ ♦♥ L 2 (N ) ✐t s✉✜❝❡s t♦ ♣r♦✈❡ t❤❛t g (d) 2 2 t f 1 ✳ ◆♦t✐❝❡ t❤❛t g (d) ✐s ♣♦s✐t✐✈❡✳ s,k ✳ Pr♦♦❢✳ ❚❤✐s ✐s ♦❜t❛✐♥❡❞ ❜② ❛♥ ❆❜❡•✬s tr❛♥s❢♦r♠✳ ❋✐rst✱ ✜① j 0 ✿ i<j≤j0 p i f j p j = i<j≤j0 p i f j (q j-1q j ) = i≤j<j0 p i df j+1 q j -i<j0 p i f j0 q j0 = i≤j<j0 p i df j+1 q jq ⊥ j0-1 f j0 q j0 ▲❡tt✐♥❣ j 0 ❣♦ t♦ ✐♥✜♥✐t②✱ ✇❡ ♦❜t❛✐♥✿ i<j∈Z p i f j p j = i≤j∈Z p i df j+1 q jq ⊥ f q✱ 2 ) ≤ 2τ (p k f k+s q k+s-1 f k+s p k ) + 2τ (p k f k+s-1 q k+s-1 f k+s-1 p k )

g (d) 1 = i∈Z τ (p i f i p i ) = i∈Z τ (p i f ) = τ ( i∈Z p i f ) = f 1 ✳ ❇② ♦rt❤♦❣♦♥❛•✐t②✱ ✇❡ ❤❛✈❡ g (d) ∞ = sup k∈Z p k f k p k ≤ 2 n t✳ ■♥❞❡❡❞✱ ❢♦r k < ∞✱ ✇❡ ❤❛✈❡✱ ❜② ▲❡♠♠❛ 4.3.3✿ p k f k p k ≤ 2 n p k f k-1 p k ≤ 2 n q k-1 f k-1 q k-1 ≤ 2 n t ❋♦r k = ∞✱ r❡❛s♦♥✐♥❣ ✐♥ L 2 (N )✿ t -qf q = lim k→∞ t -qf k q ≥ 0 s✐♥❝❡ L 2 (N ) + ✐s ❝•♦s❡❞✳ ❙♦ g d ∞ ≤ 2 n t✳ ❲❡ ❝♦♥❝•✉❞❡ ❜② ❍ö•❞❡r✬s ✐♥❡q✉❛•✐t②✿
❇② ❈✉❝✉•❡s❝✉✬s t❤❡♦r❡♠ ❛♥❞ r❡❝❛••✐♥❣ t❤❛t f i ≤ 2 n f i-1 ❢r♦♠ •❡♠♠❛ 4.3.3✿ f 1/2 k+s q k+s-1 f 1/2 k+s ∞ = q k+s-1 f k+s q k+s-1 ∞ t f 1/2 k+s q k+s-1 f 1/2 k+s-1 ∞ = q k+s-1 f k+s-1 q k+s-1 ∞ ≤ t ❚❤❡r❡❢♦r❡✱ ❢♦r ❛•• s > 0✿ g (l) s 2 2 = k>0 g (l) k,s
M A = M⊗A✱ N A = N ⊗A ❛♥❞ t❤❡ ✐♥❝•✉s✐♦♥ ♠❛♣s i 1 : N → N A ❛♥❞ i 2 : ℓ 2 → M A ❜②✿ i 1 (x) = 1 ⊗ x ❛♥❞ i 2 (h) = ∞ k=0 h k ⊗ λ(g k )✳
-m ❛♥❞ N = N ⊗L ∞ (Ω)✳ ❈♦♥s✐❞❡r N ❛s ❛ s✉❜❛•❣❡❜r❛ ♦❢ N ✳ ❲❡ ✇✐•• t❛❦❡ ❛ ❢✉♥❝t✐♦♥❛• ❛♣♣r♦❛❝❤ ❛♥❞ ✇♦r❦ ✐♥ N ❢r♦♠ ♥♦✇ ♦♥✳ ❉❡♥♦t❡ ❜② ε Q t❤❡ Q✲❝♦♦r❞✐♥❛t❡ ✐♥ Ω✳ ❉❡✜♥❡ g m (x, ω) = Q∈Qm I Q (x)ε Q (ω)✱ l m (x, ω) = Q∈Qm I 5Q (x)ε Q (ω)✳ ▲❡t G m ✭r❡s♣✳ L m ✮ ❜❡ t❤❡ ♠✉•t✐♣•✐❝❛t✐♦♥ ❜② g m ✭r❡s♣✳ l m ✮ ❛♥❞ E ❜❡ t❤❡ ❝♦♥❞✐t✐♦♥♥❛• ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ L p (N )✳ ◆♦t❡ t❤❛t
Φ i = k∈Z ∆ k-s+i T k-s ∆ k = k∈Z ∆ k-s+i T -EL k-s T G k-s ∆ k ✱ ✇❤❡r❡ t❤❡ L k ❛♥❞ G k ❛r❡ ❣✐✈❡♥ ❜② t❤❡ •❡♠♠❛ ❛❜♦✈❡✳ ❲❡ ❝❛♥ ✇r✐t❡ Φ i = ∆T - ∆ELT G ✇❤❡r❡ Gx = (G k-s x k ) k∈Z ✱ Lx = (L k-s x k ) k∈Z ✱ ∆x = (∆ k-s+i x k ) k∈Z ❛♥❞ T x = (T x k ) k∈Z ✳ L ❛♥❞ G ❛r❡ st✐•• ❜♦✉♥❞❡❞ ♠✉•t✐♣•✐❝❛t✐♦♥s ❜② ✉♥❝♦♥✲ ❞✐t✐♦♥♥❛•✐t② ♦❢ RC p ( N )✳ T ✐s ❝♦♠♣•❡t❡•② ❜♦✉♥❞❡❞ ♦♥ RC p ( N ) ❛s ❛ ❞✐❛❣♦♥❛• ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r✱ E ❛s ❛ ❝♦♥❞✐t✐♦♥❛• ❡①♣❡❝t❛t✐♦♥ ❢r♦♠ RC p ( N ) t♦ RC p (N ) ❛♥❞ ∆ ❜② ❙t❡✐♥✬s ✐♥❡q✉❛•✐t② ✭ ❚❤❡♦r❡♠ 2.3 ✐♥ ❬✹✾(✮✿ (E n (x n )) Lp( N ,ℓ c 2 ) ≤ (x n ) Lp( N ,ℓ c 2 ) ❛♥❞ (E n (x n )) Lp( N ,ℓ r 2 ) ≤ (x n ) Lp( N ,ℓ r 2 ) .

❍❡♥❝❡✱ t❤❡

Ψ i = k∈Z ∆ k-i S k,s ∆ k+s = k∈Z ∆ k-i (T k,s + R k,s )∆ k+s = A i + B i ✇❤❡r❡ A i = k∈Z ∆ k-i T k,s ∆ k+s = k∈Z ∆ k-i T k ∆ k+s ✭❜② ✹✳✻✮ ❛♥❞ B i = k∈Z ∆ k-i R k,
U λ ❣♦❡s t♦ -1 ✇❤❡♥ λ ❣♦❡s t♦ ∞ ❛♥❞ t♦ 1 ✇❤❡♥ λ = 0 s♦ t❤❡r❡ ♠✉st ❡①✐st ❛ s✉✐t❛❜•❡ U λ ✳ ❊①t❡♥❞ U t♦ R n ❜② U (0) = 0 ❛♥❞ U (x) = U (x/ |x|) ♦t❤❡r✇✐s❡✳ ❲r✐t❡✱

  r❡♥t③✲❙❤✐♠♦❣❛❦✐ ❣é♥ér❛•✐sé ♣❡r♠❡t ❞❡ •✬ét❡♥❞r❡ à t♦✉s •❡s ❡s♣❛❝❡s ✐♥t❡r♣♦•és ❡♥tr❡ L p ❡t L ∞ ✳ ❉❛♥s •❡ ❝❛s ♣❛rt✐❝✉•✐❡r ❞❡ L 1 ✱ ♦♥ ♣❡✉t ♠♦♥tr❡r q✉✬✐• ❡①✐st❡ t♦✉•♦✉rs ❞❡s ❞é❝♦♠♣♦s✐t✐♦♥s ♦♣t✐♠❛•❡s✳ ❈❡•❛ s✐♠♣•✐✜❡ •❛ ♣r❡✉✈❡ ❞✉ t❤é♦rè♠❡ ♣ré❝é❞❡♥t ❡t ♣❡r♠❡t ❛✉ss✐ ❞❡ ♣♦✉ss❡r ♣•✉s •♦✐♥ •✬ét✉❞❡ ❞❡ ❝❡s ❞é❝♦♠♣♦s✐t✐♦♥s ❡t ❞✬♦❜t❡♥✐r ✉♥❡ ❝❡rt❛✐♥❡ ✐♥❞❡♥t✐té ❛•❣é❜r✐q✉❡ q✉✐ •❡s ❝❛r❛❝tér✐s❡✳ ❆ •✬❛✐❞❡ ❞❡ ❝❡tt❡ ❞❡r♥✐èr❡✱ q✉✐ s❡♠❜•❡ •♦✉❡r •❡ rô•❡ ❞✬✉♥❡ ❞é❝♦♠♣♦s✐t✐♦♥ ♣♦•❛✐r❡ ♣♦✉r •❡s s✉✐t❡s✱ ♥♦✉s r❡tr♦✉✈♦♥s •❛ ♣•✉♣❛rt ❞❡s rés✉•t❛ts ❝♦♥♥✉s s✉r •❡s ✐♥é❣❛•✐tés ❞❡ ❑❤✐♥t❝❤✐♥❡ ❞❛♥s •❡ ❝❛s •✐❜r❡✳ ❚❤é♦rè♠❡ ✵✳✶✳✶✵✳ ❙♦✐t

  r❡✉✈❡ ❞✉ t❤é♦rè♠❡ s✬❡✛❡❝t✉❡ ❡♥ ❞❡✉① ét❛♣❡s✳ Pr❡♠✐èr❡♠❡♥t✱ •✬✐❞❡♥t✐té ❛•❣é❜r✐q✉❡ ❡st ♦❜t❡♥✉❡ ♣❛r ✉♥ ❛r❣✉♠❡♥t ❞❡ ❞✉❛•✐té✳ ❖♥ ❝♦♥s✐❞ér❡ ✉♥❡ ❞é❝♦♠✲ ♣♦s✐t✐♦♥ ♦♣t✐♠❛•❡ x = y + z ❡t •✬♦♥ é❝r✐t •❡s ❞é❝♦♠♣♦s✐t✐♦♥s ♣♦•❛✐r❡s ❞❡ y ❡t z✳ ▲✬♦♣t✐♠❛•✐té s❡ tr❛❞✉✐t ♣❛r •❡ ❢❛✐t q✉❡ ❝❡s ❞é❝♦♠♣♦s✐t✐♦♥s s❛t✉r❡♥t ❝❡rt❛✐♥❡s ✐♥é❣❛•✐tés✱ ❡t ❞♦♥❝ ✈ér✐✜❡♥t ✉♥❡ r❡•❛t✐♦♥ q✉✐ ❞♦♥♥❡ •✬✐❞❡♥t✐té ❛tt❡♥❞✉❡✳ ❉❛♥s ✉♥ s❡❝♦♥❞ t❡♠♣s✱ ♦♥ ✉t✐•✐s❡ ❝❡tt❡ ✐❞❡♥t✐té ♣♦✉r ♦❜t❡♥✐r ✉♥❡ ♥♦✉✈❡••❡ ❢♦✐s ❞❡s ♠❛•♦r❛t✐♦♥s ❡♥ t❡r♠❡ ❞❡ K✲❢♦♥❝t✐♦♥♥❡••❡✱ q✉✐ ✐♠♣•✐q✉❡♥t •❡ t❤é♦rè♠❡ ❝✐✲❞❡ss✉s✳ ◆♦t♦♥s q✉❡ ♣♦✉r p ≥ 1✱ •❡s ❝❛•❝✉•s s♦♥t é•é♠❡♥t❛✐r❡s ♠❛✐s ♣♦✉r p < 1✱ ♥♦✉s ❞❡✲ ✈♦♥s ✉t✐•✐s❡r ✉♥❡ ✐♥é❣❛•✐té ❞❡ ❍ö•❞❡r ♣♦✉r •❡s ❝♦♠♠✉t❛t❡✉rs✱ ♣r♦✉✈é❡ ré❝❡♠♠❡♥t ♣❛r |✐❝❛r❞ ❬✺✹(✳ ■• s✬❛❣✐t ❞✬✉♥ rés✉•t❛t t❡❝❤♥✐q✉❡ très s✐♠✐•❛✐r❡ à ❝❡•✉✐ ✉t✐•✐sé ♣❛r P✐s✐❡r ❡t |✐❝❛r❞ ❞❛♥s ❬✹✽( ♣♦✉r ♦❜t❡♥✐r •❡s ✐♥é❣❛•✐tés ❞❡ ❑❤✐♥t❝❤✐♥❡ ♣♦✉r p < 1✳

  ✉s •❡s ✐♥t❡r♣♦•és ❞✬❡s♣❛❝❡s L p ✳ ❊♥ ♣❛rt✐❝✉•✐❡r✱ •❡s ❝❛s p < 2 ❡t p > 2 s♦♥t tr❛✐tés ❞❡ •❛ ♠ê♠❡ ❢❛ç♦♥✱ ❝❡ q✉✐ ❡st ✉♥❡ ♥♦✉✈❡❛✉té ❞❛♥s •✬ét✉❞❡ ❞❡s ✐♥é❣❛•✐tés ❞❡ ❑❤✐♥t❝❤✐♥❡ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡s✳ ❋✐♥❛•❡♠❡♥t✱ ♥♦✉s ❞♦♥♥♦♥s✱ s♦✉s ❞❡s ❝♦♥❞✐t✐♦♥s t❡❝❤♥✐q✉❡s s✉r •❡s ❡s♣❛❝❡s ❝♦♥s✐❞érés ❡t ❞❛♥s •❡ ❝❛❞r❡ ❞❡s ✉♥✐t❛✐r❡s ❞❡ ❍❛❛r •✐❜r❡s✱ ✉♥❡ ❝❤❛r❛❝tér✐s❛t✐♦♥ ❞❡s ❡s♣❛❝❡s ❞❛♥s •❡sq✉❡•s •❡s ❢♦r♠✉•❡s ✉s✉❡••❡s ♣♦✉r •❡s ✐♥é❣❛•✐tés ❞❡ ❑❤✐♥t❝❤✐♥❡ s♦♥t ✈❛•✐❞❡s✳ ❚❤é

  ❡t k : R 2n → C ❡st ❛♣♣❡•é •❡ ♥♦②❛✉ ❞❡ T ❡t ✈ér✐✜❡ ❞❡s ❝♦♥❞✐t✐♦♥s ✭q✉❡ ♥♦✉s ❛♣♣❡••❡r♦♥s ✧❝♦♥❞✐t✐♦♥s ❞❡ ❈❛•❞❡ró♥✲)②❣♠✉♥❞✧ ❜✐❡♥ q✉❡ ❝❡tt❡ t❡r♠✐♥♦•♦❣✐❡ ♥❡ s♦✐t ♣❛s st❛♥❞❛r❞✮ q✉✐ ♣❡r♠❡tt❡♥t ❞❡ ❝♦♥trô•❡r •❡s ✈❛r✐❛✲ t✐♦♥s ❞❡ k ♠❛✐s •❛✐ss❡♥t •❛ ♣♦ss✐❜✐•✐té ❞✬❛✈♦✐r ✉♥❡ s✐♥❣✉•❛r✐té s✉r •❛ ❞✐❛❣♦♥❛•❡✳ ❯♥❡ s♦✉r❝❡ ❞✬❡①❡♠♣•❡s ❞❡ t❡•s ♥♦②❛✉① ❡st ❞♦♥♥é❡ ♣❛r •❡s ♥♦②❛✉① ❞❡ ❝♦♥✈♦•✉✲ t✐♦♥✱ ❝✬❡st✲à✲❞✐r❡ ❝❡✉① ♣♦✉r •❡sq✉❡•s ✐• ❡①✐st❡ ✉♥❡ ❢♦♥❝t✐♦♥ K : R n → C t❡••❡ q✉❡ k(x, y) = K(xy)✳ ❉❛♥s ❝❡ ❝❛s✱ ✉♥❡ ❝♦♥❞✐t✐♦♥ ❞❡ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ t②♣✐q✉❡ ❡st ✿ |∇K(x)| 1 |x| n+1 . ❈❡tt❡ ❝♦♥❞✐t✐♦♥ ❡st r❛r❡♠❡♥t ♦♣t✐♠❛•❡ ♠❛✐s ❢❛❝✐•❡ à ✈ér✐✜❡r ❡♥ ♣r❛t✐q✉❡✳ ❊••❡ ❡st ✈ér✐✜é❡ ♣❛r •❡s ♥♦②❛✉① ❞❡ •❛ tr❛♥s❢♦r♠é❡ ❞❡ |✐❡s③ ❡t ❞❡ •❛ tr❛♥s❢♦r♠é❡ ❞❡ ❍✐•❜❡rt✱ q✉✐ s♦♥t ❞❡s ❡①❡♠♣•❡s ❢♦♥❞❛♠❡♥t❛✉① ❞❡ •❛ t❤é♦r✐❡ ❞❡s ✐♥té❣r❛•❡s s✐♥❣✉•✐èr❡s✳ ◆♦✉s s♦♠♠❡s ✐♥tér❡ssés ♣❛r •❡s ♣♦t❡♥t✐❡••❡s ❣é♥ér❛•✐s❛t✐♦♥s ♥♦♥✲❝♦♠♠✉t❛t✐✈❡s ❞✉ t❤é♦rè♠❡ s✉✐✈❛♥t✱ q✉❡ ♥♦✉s ❝✐t♦♥s ❞❡ ❢❛ç♦

  ❝❤❛q✉❡ ❡s♣❛❝❡ ❞❡ ❇❛♥❛❝❤ X✳ ▲❡s ❝♦♥❞✐t✐♦♥s q✉✬✐• ❢❛✉t ✐♠♣♦s❡r s✉r X ♣♦✉r q✉❡•❛ ❜♦r♥✐t✉❞❡ ❞❡ T s♦✐t ❛•♦rs ❝♦♥s❡r✈é❡ ♦♥t été ét✉❞✐é❡✱ •❛ ❝♦♥❞✐t✐♦♥ ❝•é ét❛♥t q✉❡ X s♦✐t ❯▼❉✳ ▲❡s ❡s♣❛❝❡s L p ♥♦♥✲❝♦♠♠✉t❛t✐❢s s♦♥t •✉st❡♠❡♥t ❯▼❉ ♣♦✉r p ∈ (0, ∞) ❡t ❞♦♥❝ •❡s ✐♥té❣r❛•❡s s✐♥❣✉•✐èr❡s ♣❡✉✈❡♥t ② êtr❡ ✉t✐•✐sé❡s✳ ❈❡♣❡♥❞❛♥t✱ •❛ t❤é♦r✐❡ ❞❡s ♦♣ér❛t❡✉rs ❞❡ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ à ✈❛•❡✉rs ✈❡❝t♦r✐❡••❡s ♥❡ ♣r♦✲ ❞✉✐t ♣❛s ✉♥ ❛♥❛•♦❣✉❡ ♥♦♥✲❝♦♠♠✉t❛t✐❢ ❞✉ t❤é♦rè♠❡ ❞❡ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ q✉✐ ♣❡r♠❡ttr❛✐t ❞✬♦❜t❡♥✐r ❞❡s ❜♦r♥❡s ♦♣t✐♠❛•❡s ♣♦✉r •❛ ♥♦r♠❡ ❞❡ T ❞❛♥s L p ✈❡rs L p •♦rsq✉❡ p t❡♥❞ ✈❡rs 1✳ ❉❛♥s ❬✹✶(✱ P❛r❝❡t ❛ ❣é♥ér❛•✐sé •❡ t❤é♦rè♠❡ ✵✳✶✳✶✷ ❛✉ ❝❛❞r❡ ♥♦♥✲❝♦♠♠✉t❛t✐❢✳ ◆♦t♦♥s q✉❡ ❞❛♥s •❡ ❝❛❞r❡ ♥♦♥✲❝♦♠♠✉t❛t✐❢✱ •❡s ♠ét❤✲ ♦❞❡s ❞✐r❡❝t❡s ♦✉ ❞✬✐♥t❡r♣♦•❛t✐♦♥ ❢❛✐s❛♥t ✐♥t❡r✈❡♥✐r •❡s ❡s♣❛❝❡s ❇▼❖ ♣❡✉✈❡♥t êtr❡ ♣ré❢éré❡s ❛✉① ✐♥é❣❛•✐tés ❞❡ t②♣❡ ❢❛✐❜•❡ q✉✐ s♦♥t ♣❛r❢♦✐s t❡❝❤♥✐q✉❡s✳ ❚♦✉t❡❢♦✐s✱ •❡ rés✉•t❛t ❞❡ P❛r❝❡t ❛ tr♦✉✈é ❞❡ ♥♦♠❜r❡✉s❡s ❛

f

  Lip := sup t =s |f (t)f (s)| |t -s| . |é❝❡♠♠❡♥t✱ •❡s ❞❡✉① ❛✉t❡✉rs✱ ❞❛♥s ✉♥❡ ❝♦••❛❜♦r❛t✐♦♥ ❛✈❡❝ ❈❛s♣❡rs ❡t )❛♥✐♥✱ ♦♥t ❞♦♥♥é ✉♥❡ ♥♦✉✈❡••❡ ♣r❡✉✈❡ ❞❡ ✭✷✮ q✉✐ ♣❛ss❡ ♣❛r ✉♥❡ ✐♥é❣❛•✐té ❞❡ t②♣❡ ❢❛✐❜•❡ ❡t ❞é❝r✐t •❡ ❝♦♠♣♦rt❡♠❡♥t ♦♣t✐♠❛• ❞❡ •❛ ❝♦♥st❛♥t❡ c p q✉❛♥❞ p t❡♥❞ ✈❡rs 1✳ ❊••❡ ❡st ❜❛sé❡ s✉r •❛ t❤é♦r✐❡ ❞❡ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ❡t ❡♥ ♣❛rt✐❝✉•✐❡r s✉r •❡ t❤é♦rè♠❡ ❞❡ P❛r❝❡t✳ ▲❡ s❝❤é♠❛ ❞❡ ♣r❡✉✈❡ ❡♠♣•♦②é ✲ ❞❡ ré❞✉✐r❡ ✉♥ ♣r♦❜•è♠❡ à •❛ ❜♦r♥✐✲ t✉❞❡ ❞✬✉♥ ♦♣ér❛t❡✉r ❞❡ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ✲ ♣♦✉rr❛✐t êtr❡ ❛♣♣•✐q✉é❡ à ❞✬❛✉tr❡s q✉❡st✐♦♥s✳ ❈✬❡st •❛ r❛✐s♦♥ ♣r✐♥❝✐♣❛•❡ ♣♦✉r •❛q✉❡••❡ •❡ t❤é♦rè♠❡ ❞❡ P❛r❝❡t r❡ç♦✐t ❝❡s ❞❡r♥✐èr❡s ❛♥♥é❡s ✉♥❡ ❛tt❡♥t✐♦♥ ✐♠♣♦rt❛♥t❡✳ ❈❡♣❡♥❞❛♥t✱ •❛ ❞é♠♦♥str❛t✐♦♥ ❞♦♥♥é❡ ♣❛r P❛r❝❡t ❞❛♥s ❬✹✶( ❡st ❞✐✣❝✐•❡ ❡t ♣❛r ❝♦♥séq✉❡♥t ❞✐✣❝✐•❡ à ❛❞❛♣t❡r s✐ •✬♦♥ r❡♥❝♦♥tr❡ ✉♥❡ s✐t✉❛t✐♦♥ ❞❛♥s •❛q✉❡••❡ •❡ t❤é♦rè♠❡ ♥❡ s✬❛♣♣•✐q✉❡ ♣❛s ✐♠♠é✲ ❞✐❛t❡♠❡♥t✳ ❈❡❝✐ ❛ ♠♦t✐✈é ♥♦tr❡ tr❛✈❛✐•✳ ◆♦✉s ♣rés❡♥t♦♥s ✉♥❡ ♣r❡✉✈❡ s✐♠♣•✐✜é❡ ❞❡ •✬✐♥é❣❛•✐té ❞❡ t②♣❡ ❢❛✐❜•❡ ♣♦✉r •❡s ✐♥té❣r❛•❡s s✐♥❣✉•✐èr❡s ♥♦♥❝♦♠♠✉t❛t✐✈❡s✱ ❡♥ s✬❛♣♣✉②❛♥t s✉r •❡s ♠ê♠❡s ✐❞é❡s q✉❡ P❛r❝❡t✳ ❈♦♠♠❡♥ç♦♥s ♣❛r ❞♦♥♥❡r ✉♥ é♥♦♥❝é ♣ré❝✐s ❞✉ t❤é♦rè♠❡✳ ▲❡s ❝♦♥❞✐t✐♦♥s ❞❡ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♥é❝❡ss❛✐r❡s ♣♦✉r ❢❛✐r❡ ❢♦♥❝t✐♦♥♥❡r •❛ ❞é♠♦♥str❛t✐♦♥ ❞❛♥s •❡ ❝❛❞r❡ ♥♦♥✲❝♦♠♠✉t❛t✐❢ s♦♥t •❡s s✉✐✈❛♥t❡s ✿ ❉é✜♥✐t✐♦♥ ✵✳✶✳✺ ✭▲✐ss✐té ❡t ❚❛✐••❡✮✳ ❖♥ ❞✐t q✉❡ T ❛ ♣♦✉r ♣❛r❛♠ètr❡ ❞❡ ▲✐♣s❝❤✐t③ γ ✭0 < γ ≤ 1✮ s✐ ♣♦✉r t♦✉t x✱y ❡t z ❞❛♥s R n t❡•s q✉❡ |x -z| ≤1 2 |y -z|✱ •❡s ✐♥é❣❛•✐tés ❞❡ •✐ss✐té s✉✐✈❛♥t❡s s♦♥t ✈ér✐✜é❡s ✿ |k(x, y)k(z, y)| |x -z| γ |y -z| n+γ , |k(y, x)k(y, z)| |x -z| γ |y -z| n+γ .

f 1 f 1 .

 1 y ∈ R n s✉❝❤ t❤❛t y ∈ 5Q x,i∧j ✿ ζ(x)b i,j (y)ζ(x) = 0✳ Pr♦♦❢✳ ❚❤❡ ✜rst ♣♦✐♥t ✐s str❛✐❣❤t❢♦r✇❛r❞✳ ❙✐♥❝❡ Q ∈ Q i∨j ✿ Q b i,j = Q E i∨j (b i,j ) = 0✳ ❚❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ζ ❡①♣r❡ss❡❞ ✐♥ ♣r♦♣✲ ❡rt② 4.3.4✳ ■♥❞❡❡❞✱ ❢♦r ❛•• x, y ∈ R n ❛♥❞ k ∈ Z s✉❝❤ t❤❛t y ∈ 5Q x,k ✇❡ ❦♥♦✇ t❤❛t ζ(x)p k (y) = p k (y)ζ(x) = 0✳ |❡❝❛•• t❤❛t b i,j = p i (ff i∨j )p j ✱ ✐t ✐s ♥♦✇ ❝•❡❛r t❤❛t ζ(x)b i,j (y)ζ(x) = 0 ❢♦r y ∈ 5Q x,k ❛♥❞ k = i, j ✇❤✐❝❤ ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ •❡♠♠❛✳ ◆♦t❡ t❤❛t Q x,k ⊂ Q x,k ′ ❢♦r k ≥ k ′ ✱ s♦ ✇❡ ❞♦ ♥♦t •♦s❡ ❛♥②t❤✐♥❣ ❜② t❛❦✐♥❣ k = i ∧ j✳ ❚❤❡ ❢♦••♦✇✐♥❣ •❡♠♠❛ ✐s t❤❡ ❝♦r❡ ♦❢ t❤❡ ❜❛❞ ❢✉♥❝t✐♦♥ ❡st✐♠❛t❡✱ ✐t r❡•✐❡s ♦♥ ❛ ❝♦♠♣✉t❛t✐♦♥ ✇❤✐❝❤ ❛••♦✇s ✉s t♦ ♠❛❦❡ ✉s❡ ♦❢ t❤❡ s♠♦♦t❤♥❡ss ❝♦♥❞✐t✐♦♥✳ ▲❡♠♠❛ ✹✳✸✳✾✳ ❋♦r ❛•• i, j ∈ Z ✿ ζT b i,j ζ 1 2 -|i-j|γ b i,j 1 ✳ Pr♦♦❢✳ ◆♦t❡ t❤❛t b i,j ✐s ✐♥ L 2 (N )✳ ❋✐① i, j ∈ Z ❛♥❞ x ∈ R n ✱ r❡❝❛•• t❤❛t k(x, y) ❝♦♠♠✉t❡s ✇✐t❤ ζ(x) s✐♥❝❡ ζ t❛❦❡s ✈❛•✉❡s ✐♥ M ❛♥❞ k ✐♥ M ′ ✿ ζ(x)T b i,j (x)ζ(x) = y∈R n k(x, y)ζ(x)b i,j (y)ζ(x)dy = y∈5Q c x,i∧j k(x, y)k(x, c y,i∨j ) ζ(x)b i,j (y)ζ(x)dy ✾✽❈❍❆P❚❊| ✹✳ ❲❊❆❑ ❚(P❊ ■◆❊◗❯❆▲■❚( ❋❖| ❙■◆•❯▲❆| ■◆❚❊•|❆▲❙ ✇❤❡r❡ ✇❡ ✉s❡❞ ❜♦t❤ ❝❛♥❝❡••❛t✐♦♥ ♣r♦♣❡rt✐❡s ♦❢ b i,j ✭•❡♠♠❛ 4.3.8✮ t❤❡ ✜rst ♦♥❡ t♦ ♠❛❦❡ t❤❡ t❡r♠ k(x, c y,i∨j ) ❛♣♣❡❛r ❛♥❞ t❤❡ s❡❝♦♥❞ ♦♥❡ t♦ r❡❞✉❝❡ t❤❡ ❞♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ t❤❡ s♠♦♦t❤♥❡ss ❝♦♥❞✐t✐♦♥ ✭❞❡✜♥✐t✐♦♥ ✹✳✶✳✸✮ ❛♣♣•✐❡s ❛♥❞ ❣✐✈❡s ✿ ζ(x)T b i,j (x)ζ(x) 1 ≤ y∈5Q c x,i∧j k(x, y)k(x, c y,i∨j ) M b i,j (y) 1 dy ≤ y∈R n I y / ∈5Qx,i∧j 2 -γ(i∨j) |x -y| n+γ b i,j (y) 1 dy ■t ❢♦••♦✇s t❤❛t ✿ ζT b i,j ζ 1 = x∈R n ζ(x)T b i,j (x)ζ(x) 1 dx ≤ y∈R n x∈R n I x / ∈5Qy,i∧j 2 -γ(i∨j) |x -y| n+γ b i,j (y) 1 dxdy y∈R n 2 -γ(i∨j-i∧j) b i,j (y) 1 dy 2 -γ|i-j| b i,j 1 Pr♦♦❢ ♦❢ ♣r♦♣❡rt② 4.3.6✳ ❚❤❡ ♦♥•② t❤✐♥❣ •❡❢t t♦ ❞♦ ✐s t♦ ❣•✉❡ t❤❡ ♣✐❡❝❡s t♦❣❡t❤❡r t❤❛♥❦s t♦ •❡♠♠❛s 4.3.7 ❛♥❞ 4.3.9✿ ζT bζ 1 ≤ i,j∈Z ζT b i,j ζ 1 ≤ ✹✳✸✳✸ ❊st✐♠❛t❡ ❢♦r t❤❡ ❣♦♦❞ ❢✉♥❝t✐♦♥ ❚❤✐s ♦♥❡ ✐s ♠♦r❡ ✐♥✈♦•✈❡❞ ❛♥❞ r❡q✉✐r❡s t❤❡ L 2 ✲♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ t❤❡♦r❡♠✳ ❚❤❡ s❛♠❡ tr✐❝❦ ❛s ❢♦r t❤❡ ❜❛❞ ❢✉♥❝t✐♦♥ ❛••♦✇s ✉s t♦ ✇r✐t❡✿ λ t (T g) τ (1ζ) + λ t (ζT gζ) t❤❡ ❞✐❛❣♦♥❛•✱ •❡❢t ❛♥❞ r✐❣❤t ♣❛rts ♦❢ g ❛s ❢♦••♦✇s✿

gt f 1 ✱

 1 (r) = i<j∈Z p j f j p i + qf q ⊥ ✳ ◆♦t❡ t❤❛t t❤❡ ❡st✐♠❛t❡ ❢♦r t❤❡ ❣♦♦❞ ❢✉♥❝t✐♦♥ ❝❛♥ ❡❛s✐•② ❜❡ ❞❡❞✉❝❡❞ ❢r♦♠ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt②✿ Pr♦♣♦s✐t✐♦♥ ✹✳✸✳✶✵✳ ❚❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡s ❤♦•❞ ✿ T g (d) 2 2 ζT g (l) 2 2 t f 1 ❛♥❞ T g (r) ζ

g d 2 2

 2 ≤ g d 1 g d ∞ ≤ 2 n t f 1 ✳ Pr♦♦❢ ♦❢ t❤❡ g (l) ❡st✐♠❛t❡✳ ❚❤✐s ✇✐•• ❝♦♥❝•✉❞❡ t❤❡ ♣r♦♦❢ ♦❢ ♣r♦♣♦s✐t✐♦♥ 4.3.10 s✐♥❝❡ t❤❡ ❛r❣✉♠❡♥t ❢♦r g (r) ✐s s✐♠✐•❛r✳ ▲❡♠♠❛ ✹✳✸✳✶✶✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❡①♣r❡ss✐♦♥ ❢♦r g (l) ✿ g (l) = ∞ k=1 ∞s=1 p k df k+s q k+s-1 =:

p k df k+s q k+s- 1 ❚❤❡2 -γs t f 1 ✳ 2 t f 1 ✳ 1 22

 11211 ✇❤✐❝❤ ✐s ❡①❛❝t•② ✇❤❛t ✇❡ ♥❡❡❞❡❞✳Pr♦♣♦s✐t✐♦♥ ✹✳✸✳✶✷✳ ❉❡✜♥❡✱ ❢♦r ❛•• s ≥ 1✿ g ❚❤✐s ✐s✱ ❛s ❢♦r t❤❡ ❜❛❞ ❢✉♥❝t✐♦♥✱ t❤❡ ❝♦r❡ ♦❢ t❤❡ ♣r♦♦❢ ❛♥❞ ✇✐•• r❡q✉✐r❡ s♦♠❡ ✇♦r❦✳ ▲❡♠♠❛ ✹✳✸✳✶✸✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ✶✵✵❈❍❆P❚❊| ✹✳ ❲❊❆❑ ❚(P❊ ■◆❊◗❯❆▲■❚( ❋❖| ❙■◆•❯▲❆| ■◆❚❊•|❆▲❙ ✶✳ ∆ k+s (g (l) s ) = g (l) s,k ✳ ✷✳ ❋✐① s✱ t❤❡ g (l) s,k ❛r❡ ♦rt❤♦❣♦♥❛• ✐♥ L 2 (N )✳ ✸✳ g (l) s 2 Pr♦♦❢✳ 1 ✐s str❛✐❣❤t❢♦r✇❛r❞ ❛♥❞ 2 ❢♦••♦✇s ❞✐r❡❝t•② ❢r♦♠ 1 s✐♥❝❡ ♠❛rt✐♥❣❛•❡ ❞✐✛❡r❡♥❝❡s ❛r❡ ❛•✇❛②s ♦rt❤♦❣♦♥❛•✳ ▲❡t ✉s ♣r♦✈❡ 3✳g p k f k+s q k+s-+ p k f k+s-1 q k+s-1 2

k>0 1 ✇❤✐❝❤

 1 tτ (p k f ) = t f 1 ✳ ▲❡♠♠❛ ✹✳✸✳✶✹✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❡st✐♠❛t❡✿ ζT g Pr♦♦❢✳ ❚❤✐s ✐s ✇❤❡r❡ ✇❡ ❛♣♣•② ♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ ✐✳❡ t❤❡♦r❡♠ 4.2.1 t♦ g (l) s ✳ ❯s✐♥❣ t❤❡ ♥♦t❛t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❢♦r t❤✐s t❤❡♦r❡♠✱ ✇❡ ❝❛♥ t❛❦❡ A k = p k s✐♥❝❡ p ⊥ k g (l) s,k = 0✳ ❚❤❡♥✿ A f,s = k>0 5p k = ζ ⊥ ✳ ❚❤❡ t❤❡♦r❡♠ ❣✐✈❡s✿ ✐s ❡①❛❝t•② t❤❡ ❡①♣❡❝t❡❞ ❡st✐♠❛t❡✳ ❚❤❡ ♣r♦♣♦s✐t✐♦♥ 4.3.12 ✐s ❝•❡❛r ❢r♦♠ t❤❡ t✇♦ ♣r❡✈✐♦✉s •❡♠♠❛s✳ ■t ❢♦••♦✇s t❤❛t ✿ ✐s t❤❡ ❡①♣❡❝t❡❞ ❡st✐♠❛t❡ ❢♦r g (l) ❛♥❞ ❝♦♥❝•✉❞❡s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠✳ ✹✳✹✳ ❆PP▲■❈❆❚■❖◆ ❚❖ ❍■▲❇❊|❚✲❱❆▲❯❊❉ ❑❊|◆❊▲❙ ✶✵✶ ✹✳✹ ❆♣♣•✐❝❛t✐♦♥ t♦ ❍✐•❜❡rt✲✈❛•✉❡❞ ❦❡r♥❡•s ▲❡t (M, τ ) ❜❡ ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♠❡❛s✉r❡ s♣❛❝❡✱ N = L ∞ (R)⊗M✳ ▲❡t T ❜❡ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ t♦ ❛ ❦❡r♥❡• k t❛❦✐♥❣ ✈❛•✉❡s ✐♥ ℓ 2 ✳ ❋♦r ❛•• 1 ≤ p ≤ ∞✱ ✇❡ ❤❛✈❡ ❛ ♠✉•t✐♣•✐❝❛t✐♦♥ ❢r♦♠ ℓ 2 × L p (M) t♦ L p (M) N ✿ ❢♦r ❛•• h = (h i ) i∈N ∈ ℓ 2 ❛♥❞ x ∈ M✱ ❞❡✜♥❡ hx = (h i x) i∈N ✳ ❚❤❛♥❦s t♦ t❤❡ ♠✉•t✐♣•✐❝❛t✐♦♥ ✇❡ ❞❡✜♥❡❞✱ T ❝❛♥ ❜❡ s❡❡♥ ❢♦r♠❛••② ❛s ❛♥ ♦♣✲ ❡r❛t♦r ❢r♦♠ L p (N ) t♦ L p (N ) N ❜② t❤❡ ✉s✉❛• ❡①♣r❡ss✐♦♥ ✿ T f (x) = R k(x, y)f (y)dy. ❲❡ ✇✐•• s❤♦✇ ✐♥ t❤✐s s❡❝t✐♦♥ t❤❛t T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L 1 (N ) t♦ R 1,∞ (N ) + C 1,∞ (N )✳ ❚♦ ❛♣♣•② t❤❡ t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ t♦ ✐♥❝•✉❞❡ M ❛♥❞ ℓ 2 ✐♥ ❛ ✈♦♥ ◆❡✉♠❛♥♥ ❛•❣❡❜r❛ s✉❝❤ t❤❛t t❤❡✐r ✐♠❛❣❡s ❝♦♠♠✉t❡✳ ▲❡t ✉s ✉s❡ t❤❡ s❡tt✐♥❣ ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✳ ▲❡t (ξ i ) i∈N ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❢r❡❡ ❍❛❛r ✉♥✐t❛r✐❡s ❞❡✜♥❡❞ ♦♥ ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♣r♦❜❛❜✐•✐t② s♣❛❝❡ A✳ ❉❡✜♥❡

Ψ

  ❋r♦♠ ♥♦✇ ♦♥ M ✇✐•• ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ M⊗1 ⊂ M A ❛♥❞ ❝❛♥ t❤✉s ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ s✉❜❛•❣❡❜r❛ ♦❢ M s✐♥❝❡ τ M ❝♦✐♥❝✐❞❡s ✇✐t❤ τ M ♦♥ M✳ ■♥ t❤✐s ❝♦♥t❡①t✱ ✇❡ ❝❛♥ ❛❣❛✐♥ ❝♦♥s✐❞❡r t❤❡ ♠❛♣ G : S(N ) → N A ❞❡✜♥❡❞ ❢♦r ❛♥② ✜♥✐t❡ s❡q✉❡♥❝❡ x = (x k ) k∈N ❜② G(x) = ∞ k=0 x k ⊗ λ(g k )✳ ◆♦t❡ t❤❛t ❢♦r ❛♥② s②♠♠❡tr✐❝ s♣❛❝❡ E ✱ G ❡①t❡♥❞s t♦ H E ❛♥❞ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥t♦ ✐ts ✐♠❛❣❡ t❤❛t ✇❡ ✇✐•• ❞❡♥♦t❡ ❜② V E ❛♥❞ i 1 ❡①t❡♥❞s t♦ E(N )✳ ◆♦t❡ t❤❛t t❤❡ ✐♠❛❣❡ ♦❢ i 2 ✐s ✐♥ M ′ ∩ M A ✳ ▲❡t k : (x, y) → i 2 (k(x, y)) ❜❡ t❤❡ ❦❡r♥❡• ♦❢ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r T : L p (N ) → L p (N A )✳ ◆♦t❡ t❤❛t✱ s✐♥❝❡ t❤❡ r❛♥❣❡ ♦❢ T ✐s ✐♥❝•✉❞❡❞ ✐♥❝•✉❞❡❞ ✐♥ t❤❡ r❛♥❣❡ ♦❢ G✱ T = G • T ❛♥❞ T = G -1 • T ✭✹✳✶✵✮ Pr♦♣♦s✐t✐♦♥ ✹✳✹✳✶✳ ▲❡t T ❜❡ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ t♦ ❛ ❦❡r♥❡• k t❛❦✐♥❣ ✈❛•✉❡s ✐♥ ℓ 2 ✳ ❙✉♣♣♦s❡ ❢✉rt❤❡r♠♦r❡ t❤❛t T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L 2 (N ) t♦ R 2 N ❛♥❞ t❤❛t k s❛t✐s✜❡s t❤❡ s♠♦♦t❤♥❡ss ❛♥❞ s✐③❡ ❝♦♥❞✐t✐♦♥s t❤❡♥ t❤❡ ♦♣❡r❛t♦r T ❞❡✜♥❡❞ ❛❜♦✈❡ ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L 1 (N ) t♦ L 1,∞ (N A )✳ Pr♦♦❢✳ ❲❡ ♦♥•② ❤❛✈❡ t♦ ❝❤❡❝❦ t❤❛t ❚❤❡♦r❡♠ 4.1.5 ❝❛♥ ❜❡ ❛♣♣•✐❡❞ t♦ T ✳ ❇② ❍❛❛❣❡r✉♣✬s ✐♥❡q✉❛•✐t② ❬✶✾(✱ ❢♦r ❛•• h ∈ ℓ 2 ✱ h ≈ i 2 (h) M s♦ k ✈❡r✐✜❡s t❤❡ s✐③❡ ❛♥❞ s♠♦♦t❤♥❡ss ❝♦♥❞✐t✐♦♥s✳ ❋✉rt❤❡r♠♦r❡✱ ❜② ✭✹✳✶✵✮✱ T ✐s ❜♦✉♥❞❡❞ ♦♥ L 2 (N )✳ ❙♦ ❛•• t❤❡ ❤②♣♦t❤❡s✐s ❛r❡ ✈❡r✐✜❡❞ ❛♥❞ T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L 1 (N ) t♦ L 1,∞ ( N )✳ ❈♦r♦••❛r② ✹✳✹✳✷✳ ▲❡t T ❜❡ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ t♦ ❛ ❦❡r♥❡• k t❛❦✐♥❣ ✈❛•✉❡s ✐♥ ℓ 2 ✳ ❙✉♣♣♦s❡ ❢✉rt❤❡r♠♦r❡ t❤❛t T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L 2 (N ) t♦ R 2 (N ) ❛♥❞ t❤❛t k s❛t✐s✜❡s t❤❡ s♠♦♦t❤♥❡ss ❛♥❞ s✐③❡ ❝♦♥❞✐t✐♦♥✳ ❚❤❡♥ T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L 1 (N ) t♦ R 1,∞ (N ) + C 1,∞ (N ) ❛♥❞ ❢r♦♠ L p (N ) t♦ R p (N ) + C p (N ) ❢♦r 1 < p < 2✳ ✶✵✷❈❍❆P❚❊| ✹✳ ❲❊❆❑ ❚(P❊ ■◆❊◗❯❆▲■❚( ❋❖| ❙■◆•❯▲❆| ■◆❚❊•|❆▲❙ Pr♦♦❢✳ ▲❡t p ∈ (1, 2)✳ ❇② r❡❛• ✐♥t❡r♣♦•❛t✐♦♥✱ s✐♥❝❡ T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L 1 (N ) t♦ L 1,∞ (N A ) ❛♥❞ ❢r♦♠ L 2 (N ) t♦ L 2 (N A )✱ T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L p (N ) t♦ L p (N A )✳ ❚❤✐s ♠❡❛♥s ❜② ✭✹✳✶✵✮ t❤❛t T ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L 1 (N ) t♦ H 1,∞ (N ) ❛♥❞ ❢r♦♠ L p (N ) t♦ H p (N )✳ |❡❝❛•• t❤❛t ❜② t❤❡ ❢r❡❡ ❑❤✐♥t❝❤✐♥❡ ✐♥❡q✉❛•✐t✐❡s ✭❬✷✵( ❛♥❞ ❝♦r♦••❛r② ✸✳✷✳✷✮✱ H 1,∞ (N ) ≈ R 1,∞ (N ) + C 1,∞ (N ) ❛♥❞ H p (N ) ≈ R p (N ) + C p (N )✱ ❝♦♥❝•✉❞✐♥❣ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❝♦r♦••❛r②✳ |❡♠❛r❦ ✹✳✹✳✸✳ T ✐s ♥♦t ❜♦✉♥❞❡❞ ❢r♦♠ L p (N ) t♦ C p (N )✱ ❢♦r 1 < p < 2✳ ■t ✐s ♥❡❝❡ss❛r② t♦ ❝♦♥s✐❞❡r R p (N ) + C p (N )✳ Pr♦♦❢✳ ❲❡ ✇✐•• ❝♦♥str✉❝t ❛ ❝♦✉♥t❡r✲❡①❛♠♣•❡ t❤❛♥❦s t♦ t❤❡ ▲✐tt•❡✇♦♦❞✲P❛•❡② ❦❡r✲ ♥❡• ✭t❤❡ s❛♠❡ ❣❡♥❡r❛• ✐❞❡❛ ❝❛♥ ❛❣❛✐♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✹✶(✮✳ ▲❡t ψ ❜❡ ❛ C ∞ ❢✉♥❝t✐♦♥ s✉♣♣♦rt❡❞ ✐♥ (1, 2)✱ ❜♦✉♥❞❡❞ ❜② 1 ❛♥❞ ❝♦♥st❛♥t ❡q✉❛• t♦ ♦♥❡ ♦♥ (5/4, 7/4)✳ ❋♦r ❛♥② i ∈ N * ✱ •❡t φ i : t → ψ(it)✳ ❈♦♥s✐❞❡r t❤❡ ❦❡r♥❡• k : R 2 → ℓ 2 ❞❡✜♥❡❞ ❜② k i (x, y) = φ i (xy) ✇❤❡r❡ φ i ❞❡♥♦t❡s t❤❡ ❋♦✉rr✐❡r tr❛♥s❢♦r♠ ♦❢ φ i ✳ ■t ✐s st❛♥❞❛r❞ t❤❛t k ✈❡r✐✜❡s t❤❡ s♠♦♦t❤♥❡ss ❡st✐♠❛t❡ ❢♦r γ = 1 ❛♥❞ t❤❡ s✐③❡ ❝♦♥❞✐t✐♦♥✳ ❚❤❡ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r T ❛ss♦❝✐❛t❡❞ t♦ k ✐s ❜♦✉♥❞❡❞ ♦♥ L 2 ❜② P•❛♥❝❤❡r❡•✬s t❤❡♦r❡♠✳ ▲❡t M = B(ℓ 2 )✳ ❆s ❛•✇❛②s✱ •❡t N = L ∞ (R) ⊗ M✳ ❋♦r ❛•• p ≥ 1✱ T ❝❛♥ ❜❡ s❡❡♥ ❛s ❛♥ ♦♣❡r❛t♦r ❢r♦♠ L p (N ) t♦ C p (N )✳ ❋♦r ❛•• k > 0✱ •❡t g k ❜❡ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ I ( 5 4 2 k-1 , 5 4 2 k-1 +1/2) ✳ ◆♦t✐❝❡ t❤❛t φ i * g k = δ✳ ❋✐① m > 0 ❛♥ ✐♥t❡❣❡r✳ ▲❡t f m = m k=1 g k ⊗ e 1,k t❤❡♥ T f m = (g k ⊗ e 1,k ) e k ∈N * ✳ ■t r❡s✉•ts t❤❛t f m Lp(N ) = m 1/2 g 1 p ❛♥❞ T f m Cp(N ) = m 1/p g 1 p ✳ ❙♦ T ✐s ♥♦t ❜♦✉♥❞❡❞ ❢♦r p < 2✳ ✹✳✺ L p ✲♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐•• r❡❝♦✈❡r ✐♥ t❤✐s ❝♦♥t❡①t t❤❡ ♠❛✐♥ r❡s✉•t ❢r♦♠ ❬✷✷( ✇❤✐❝❤ ✐s ❛♥ L p ✲✈❡rs✐♦♥ ♦❢ ♣s❡✉❞♦✲•♦❝❛•✐s❛t✐♦♥✳ ❚❤❡ ♣r♦♦❢ ❤❡❛✈✐•② r❡•✐❡s ♦♥ ❡st✐♠❛t❡s ❛♥❞ ❞❡✜♥✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ✐♥ s❡❝t✐♦♥ 2. Pr❡❝✐s❡•②✱ ✇❡ ✇✐•• s❤♦✇ t❤❡ ❢♦••♦✇✐♥❣ t❤❡♦r❡♠✱ ✉s✐♥❣ ♥♦t❛t✐♦♥s ❢r♦♠ t❤❡♦r❡♠ ✹✳✷✳✶✳ ❚❤❡♦r❡♠ ✹✳✺✳✶✳ ▲❡t T ❜❡ ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❦❡r✲ ♥❡• k ✇✐t❤ ▲✐♣s❝❤✐t③ ♣❛r❛♠❡t❡r γ✱ ✈❡r✐❢②✐♥❣ t❤❡ s✐③❡ ❝♦♥❞✐t✐♦♥✱ t❛❦✐♥❣ ✈❛•✉❡s ✐♥ M ′ ∩ M ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ L 2 (N ) t♦ L 2 ( N )✳ ❚❤❡♥ ❢♦r ❛•• 1 < p < ∞✱ s ∈ N ❛♥❞ f ∈ L p (N )✱ t❤❡r❡ ❡①✐sts θ p > 0 s✉❝❤ t❤❛t ✿ A ⊥ f,s (T f ) p 2 -γsθp f p ❛♥❞ (T f )B ⊥ f,s p 2 -γsθp f p , ✇❤❡r❡ t❤❡ ✐♠♣•✐❡❞ ❝♦♥st❛♥t ❞❡♣❡♥❞s ♦♥ p✳ ❲❡ ✇✐•• ✉s❡ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✇❡ ♦❜t❛✐♥❡❞ ✐♥ t❤❡ L 2 ✲❝❛s❡ ✐♥ ✭✹✳✾✮✿ A ⊥ f,s T f = A ⊥ i f . ❚❤❡ ❝r✉❝✐❛• r❡s✉•t✱ ♣r♦✈❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥s✱ ✐s t❤❡ ❢♦••♦✇✐♥❣ ✿ ✹✳✺✳ L P ✲P❙❊❯❉❖✲▲❖❈❆▲■❙❆❚■❖◆ ✶✵✸ Pr♦♣♦s✐t✐♦♥ ✹✳✺✳✷✳ ❋♦r ❛•• p ∈ (1, ∞) t❤❡ s❡q✉❡♥❝❡s ♦❢ ♦♣❡r❛t♦rs (Φ i ) i∈N ❛♥❞ (Ψ i ) i∈N ❛r❡ ✉♥✐❢♦r♠•② ❜♦✉♥❞❡❞ ♦♥ L p (N )✳ Pr♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠✳ ❋✐① p ∈ (1, ∞)✳ ❇② ❝♦♠♣•❡① ✐♥t❡r♣♦•❛t✐♦♥✱ t❤❡r❡ ❡①✐st q ∈ (1, ∞) ❛♥❞ θ p > 0 s✉❝❤ t❤❛t Φ i B(Lp(N )) ≤ Φ i 2θp B(L2(N )) Φ i 1-2θp B(Lq(N )) . ◆♦✇✱ ✐t s✉✣❝❡s t♦ ♣•✉❣ ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣r♦♣♦s✐t✐♦♥ ❛♥❞ ♣r♦♣♦s✐t✐♦♥ ✹✳✷✳✻ t♦ ❣❡t t❤❡ ❡①♣❡❝t❡❞ ❡st✐♠❛t❡✳ ❲❡ ♦❜t❛✐♥✿ Φ i B(Lp(N )) 2 -θp(i+s) ✳ ❆ s✐♠✐•❛r ❡st✐♠❛t❡ ✐s tr✉❡ ❢♦r Ψ i ✳ ❚❤✐s ✐s ❡♥♦✉❣❤ t♦ ❝♦♥❝•✉❞❡ t❤❛♥❦s t♦ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭✹✳✾✮ ♠❡♥t✐♦♥♥❡❞ ❛❜♦✈❡✳ ✹✳✺✳✶ L p ✲❜♦✉♥❞❡❞♥❡ss ♦❢ Φ i▲❡♠♠❛ ✹✳✺✳✸✳ ❈♦♥s✐❞❡r ❛ ❈❛•❞❡ró♥✲)②❣♠✉♥❞ ♦♣❡r❛t♦r T ✇✐t❤ ❦❡r♥❡• k✱ ❛♥❞ ❛♥ ✐♥t❡❣❡r m✳ ❚❤❡♥✱ ❢♦r ❛♥② p ∈ (1, ∞)✱ t❤❡ ❦❡r♥❡• k ′ : (x, y) → I x∈5Qy,m k(x, y) ❞❡✜♥❡s ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r T ′ ♦♥ L p (N )✳ Pr♦♦❢✳ ❲❡ ✐♥tr♦❞✉❝❡ ❛ s❡q✉❡♥❝❡ ♦❢ |❛❞❡♠❛❝❤❡r ✈❛r✐❛❜•❡s ✐♥❞❡①❡❞ ❜② t❤❡ s❡t ♦❢ ❞②❛❞✐❝ ❝✉❜❡s ♦❢ s✐③❡ 2 -m t♦ ❡①♣r❡ss t❤❡ ❦❡r♥❡• k ′ ✐♥ ❛ s✉✐t❛❜•❡ ✇❛②✳ ▲❡t Ω ❜❡ t❤❡ ♣r♦❜❛❜✐•✐t② s♣❛❝❡ {-1, 1} Qm ✇❤❡r❡ Q m ✐s t❤❡ s❡t ♦❢ ❞②❛❞✐❝ ❝✉❜❡s ♦❢ s✐③❡ 2

  G m B(Lp(N )) = g m ∞ = 1 ❛♥❞ L m B(Lp(N )) = l m ∞ = 5 n ✳ ■t ✐s ❡❛s✐•② ❝❤❡❝❦❡❞ t❤❛t T ′ = EL m T G m s✐♥❝❡ k ′ (x, y) = Ω l m (x, ω)k(x, y)g m (y, ω)dω. ❙♦ T ′ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r ♦♥ L p (N )✳ Pr♦♣♦s✐t✐♦♥ ✹✳✺✳✹✳ ▲❡t p ∈ (1, ∞)✳ ❚❤❡ ♦♣❡r❛t♦rs Φ i ❛r❡ ✉♥✐❢♦r♠•② ❜♦✉♥❞❡❞ ♦♥ L p (N )✳ Pr♦♦❢✳ ❲❡ ✐❞❡♥t✐❢② L p (N ) q✉❛s✐✲✐s♦♠❡tr✐❝❛••② ✇✐t❤ ❛ s✉❜s♣❛❝❡ ♦❢ RC p ( N )✱ t❤❛♥❦s t♦ ❇✉r❦❤♦•❞❡r✲•✉♥❞② ✐♥❡q✉❛•✐t② ✭t❤❡♦r❡♠ 2.1 ✐♥ ❬✹✾(✮✿ f p ≈ df RCp( N ) ✳ |❡❝❛•• t❤❛t

  Φ i ❛r❡ ✉♥✐❢♦r♠•② ❜♦✉♥❞❡❞✳ ✶✵✹❈❍❆P❚❊| ✹✳ ❲❊❆❑ ❚(P❊ ■◆❊◗❯❆▲■❚( ❋❖| ❙■◆•❯▲❆| ■◆❚❊•|❆▲❙ ✹✳✺✳✷ L p ✲❜♦✉♥❞❡❞♥❡ss ♦❢ Ψ i |❡❝❛•• t❤❛t✱

  s ∆ k+s ✳ ❚❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ t❡••s ✉s t❤❛t A i ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L p (N ) t♦ L p (N )✱ ✇✐t❤ ❛ ❝♦♥tr♦• ♦♥ t❤❡ ♥♦r♠ ✐♥❞❡♣❡♥❞❛♥t ❢r♦♠ i✳ ❙♦ ✇❤❛t ✐s •❡❢t t♦ ♣r♦✈❡ ✐s t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✳ Pr♦♣♦s✐t✐♦♥ ✹✳✺✳✺✳ ▲❡t p ∈ (1, ∞)✱ t❤❡♥ B i ✐s ❜♦✉♥❞❡❞ ❢r♦♠ L p (N ) t♦ L p (N )✱ ✉♥✐❢♦r♠•② ✐♥ i✳ Pr♦♦❢✳ ❯s✐♥❣ ♥♦t❛t✐♦♥s ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ ✇❡ ✇✐•• ❛❣❛✐♥ ❝♦♥s✐❞❡r B i ❛s ❛ ♣❛rt✐❛• ♦♣❡r❛t♦r ❢r♦♠ RC p ( N ) t♦ RC p ( N )✳ ▲❡t φ ❜❡ ❛ ❙❝❤✇❛r③ ❢✉♥❝t✐♦♥ ❢r♦♠ R n t♦ R s✉❝❤ t❤❛t I A0 (x, y) ≤ φ(x -y) ≤ I |x-y|>1/2 ❢♦r ❛•• x, y ∈ R n ✳ ▲❡t U ❜❡ ❛ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ❢r♦♠ S n-1 t♦ t❤❡ ✉♥✐t ❝✐r❝•❡ ♦❢ C ✇✐t❤ ❛✈❡r❛❣❡ 0✳ ❙✉❝❤ U ❡①✐sts✱ t❛❦❡ ❢♦r ❡①❛♠♣•❡ U λ (x) = e 2i arctan(λx1) ✳ ❙✐♥❝❡ arctan ✐s ❛♥ ♦❞❞ ❢✉♥❝t✐♦♥✱ t❤❡ ❛✈❡r❛❣❡ ♦❢ U λ ✐s r❡❛•✳ ◆♦✇✱ ♥♦t❡ t❤❛t

  r k,s (x, y) = I A k (x, y) K(x)2 -(k+s)γI n,γ |x -y| n+γ = K(x)2 -sγ I n,γ .I A k (x, y). U (xy)φ(2 k (xy)) |2 k (xy)| γ . U (xy) |x -y| n ❉❡♥♦t❡ ❜② M K t❤❡ ♠✉•t✐♣•✐❝❛t✐♦♥ ❜② K(.)2 -sγ I n,γ ✱ M K ✐s ❜♦✉♥❞❡❞ ✭s❡❡ s❡❝t✐♦♥ 2.3✮✳ ❉❡✜♥❡✿ F (x) = U (x)φ(x) |x| γ . ❙✐♥❝❡ φ ✐s ③❡r♦ ✐♥ ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ t❤❡ ♦r✐❣✐♥✱ F ❤❛s ♥♦ s✐♥❣✉•❛r✐t② ❛♥❞ ✐s ❛ ❙❝❤✇❛r③ ❢✉♥❝t✐♦♥✳ ❙♦ F ✐s t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ ❛♥ L 1 ❢✉♥❝t✐♦♥ F ✳ ◆♦t❡ ❛•s♦ t❤❛t I A k = I x∈5Q y,k -I x∈3Q y,k ✱ ✇❡ ✇✐•• ♦♥•② ♣r♦✈❡ t❤❡ ❜♦✉♥❞❡❞♥❡ss r❡♣•❛❝✐♥❣ I A k ❜② I x∈5Q y,k s✐♥❝❡ I x∈3Q y,k ✐s s✐♠✐•❛r✱ ❞❡♥♦t❡ t❤❡ ❛ss♦❝✐❛t❡❞ ♦♣❡r❛t♦r B ′ i ✳ ❉❡✜♥❡ ❛•s♦✿ h(x) = U (x) |x| n ❛♥❞ H t❤❡ ❝♦♥✈♦•✉t✐♦♥ ♦♣❡r❛t♦r ❛ss♦❝✐❛t❡❞ t♦ h✳ ❙✐♥❝❡ U ❤❛s 0 ❛✈❡r❛❣❡✱ H ✐s ❜♦✉♥❞❡❞ ❢r♦♠ RC p ( N ) t♦ RC p ( N ) ❢♦r ❛♥② p ∈ (1, ∞)✳ ■♥❞❡❡❞✱ H ❝❛♥t❤❡♥ ❜❡ ✇r✐tt❡♥ ❛s ❛♥ ❛✈❡r❛❣❡ ♦❢ ❞✐r❡❝t✐♦♥♥❛• ❍✐•❜❡rt ♦♣❡r❛t♦rs ✭s❡❡ ❬✶✺(✮✳ ❯s✐♥❣

✶✵✺

♥♦t❛t✐♦♥s ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ♣r♦♣♦s✐t✐♦♥ ❛♥❞ •❡♠♠❛✿