Alex Amenta

INTERPOLATION AND EMBEDDINGS OF WEIGHTED TENT SPACES

Keywords: 2010 Mathematics Subject Classification. 42B35 (Primary); 46E30 (Secondary). Keywords. Weighted tent spaces, complex interpolation, real interpolation, Hardy-Littlewood-Sobolev embeddings

Given a metric measure space X, we consider a scale of function spaces T p,q s (X), called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on X we identify some associated interpolation spaces, in particular certain real interpolation spaces. These are identified with a new scale of function spaces, which we call Z-spaces, that have recently appeared in the work of Barton and Mayboroda on elliptic boundary value problems with boundary data in Besov spaces. We also prove Hardy-Littlewood-Sobolev-type embeddings between weighted tent spaces.

Contents

Notation Acknowledgements 1. Preliminaries 1.1. Metric measure spaces 1.2. Unweighted tent spaces 1.3. Weighted tent spaces: definitions, duality, and atoms 2. Interpolation and embeddings 2.1. Complex interpolation 2.2. Real interpolation: the reflexive range 2.3. Real interpolation: the non-reflexive range 2.4. Hardy-Littlewood-Sobolev embeddings 3. Deferred proofs 3.1. T p,∞ -L ∞ estimates for cylindrically supported functions 3.2. T p,∞ atomic decomposition References

 which have had many applications in harmonic analysis and partial differential equations. In some of these applications 'weighted' tent spaces have been used implicitly. These spaces, which we denote by T p,q s , seem not to have been considered as forming a scale of function spaces in their own right until the work of Hofmann, Mayboroda, and McIntosh [14, §8.3], in which factorisation and complex interpolation theorems are obtained for these spaces.

In this article we further explore the weighted tent space scale. In the interests of generality, we consider weighted tent spaces T p,q s (X) associated with a metric measure space X, although our theorems are new even in the classical case where X = R n equipped with the Lebesgue measure. Under sufficient geometric assumptions on X (ranging from the doubling condition to the assumption that X = R n ), we uncover two previously unknown novelties of the weighted tent space scale.

First, we identify some real interpolation spaces between T p 0 ,q s 0 and T p 1 ,q s 1 whenever s 0 = s 1 . In Theorem 2.4 we prove that [START_REF] Amenta | Tent spaces over metric measure spaces under doubling and related assumptions[END_REF] (T p 0 ,q s 0 , T p 1 ,q s 1 ) θ,p θ = Z p θ ,q s θ for appropriately defined parameters, where the scale of 'Z-spaces' is defined in Definition 2.3. We require p 0 , p 1 , q > 1 in this identification, but in Theorem 2.9 we show that in the Euclidean setting the result holds for all p 0 , p 1 > 0 and q ≥ 1.

In the Euclidean setting, Z-spaces have appeared previously in the work of Barton and Mayboroda [START_REF] Barton | Layer potentials and boundary-value problems for second order elliptic operators with data in Besov spaces[END_REF]. In their notation we have Z p,q s (R n ) = L(p, ns + 1, q). Barton and Mayboroda show that these function spaces are useful in the study of elliptic boundary value problems with boundary data in Besov spaces. The connection with weighted tent spaces shown here is new.

Second, we have continuous embeddings

T p 0 ,q s 0 ֒→ T p 1 ,q s 1
whenever the parameters satisfy the relation

(2) s 1 -s 0 = 1 p 1 - 1 p 0 .
This is Theorem 2.19. Thus a kind of Hardy-Littlewood-Sobolev embedding theorem holds for the weighted tent space scale, and by analogy we are justified in referring to the parameter s in T p,q s as a regularity parameter. We also identify complex interpolation spaces between weighted tent spaces in the Banach range. This result is already well-known in the Euclidean setting, and its proof does not involve any fundamentally new arguments, but we include it here for completeness.

These results in this paper will play a crucial role in forthcoming work, in which we will use weighted tent spaces and Z-spaces to construct abstract homogeneous Hardy-Sobolev and Besov spaces associated with elliptic differential operators with rough coefficients. This will be an extension of the abstract Hardy space techniques initiated independently by Auscher, McIntosh, and Russ [START_REF] Auscher | Hardy spaces of differential forms on Riemannian manifolds[END_REF] and Hofmann and Mayboroda [START_REF] Hofmann | Hardy and BMO spaces associated to divergence form elliptic operators[END_REF].

Notation. Given a measure space (X, µ), we write L 0 (X) for the set of µ-measurable functions with values in the extended complex numbers C ∪ {±∞, ±i∞}. As usual, by a 'measurable function', we actually mean an equivalence class of measurable functions which are equal except possibly on a set of measure zero. We will say that a function f ∈ L 0 (X) is essentially supported in a subset E ⊂ X if µ{x ∈ X \ E : f (x) = 0} = 0.

A quasi-Banach space is a complete quasi-normed vector space; see for example [17, §2] for further information. If B is a quasi-Banach space, we will write the quasi-norm of B as either ||•|| B or ||• | B||, according to typographical needs. For 1 ≤ p ≤ ∞, we let p ′ denote the Hölder conjugate of p, which is defined by the relation 1 = 1 p + 1 p ′ , with 1/∞ := 0. For 0 < p, q ≤ ∞, we define the number δ p,q := 1 q -1 p , again with 1/∞ := 0. This shorthand will be used often throughout this article. We will frequently use the the identities δ p,q + δ q,r = δ p,r , δ p,q = δ q ′ ,p ′ , 1/q = δ ∞,q = δ q ′ ,1 .

As is now standard in harmonic analysis, we write a b to mean that a ≤ Cb for some unimportant constant C ≥ 1 which will generally change from line to line. We also write a c 1 ,c 2 ,... b to mean that a ≤ C(c 1 , c 2 , . . .)b.

Acknowledgements. I thank Pierre Portal and Pascal Auscher for their comments, suggestions, corrections, and support. I also thank Yi Huang, Gennady Uraltsev, and Christian Zillinger for interesting discussions regarding technical aspects of this work.

Some of this work was carried out during the Junior Trimester Program on Optimal Transportation at the Hausdorff Research Institute for Mathematics in Bonn. I thank the Hausdorff Institute for their support, as well as the participants of the trimester seminar for their comments. I also thank Lashi Bandara and the Préfecture de Bobigny for indirectly influencing my participation in this program.

Finally, I acknowledge financial support from the Australian Research Council Discovery Grant DP120103692, and also from the ANR project "Harmonic analysis at its boundaries" ANR-12-BS01-0013-01.

1. Preliminaries 1.1. Metric measure spaces. A metric measure space is a triple (X, d, µ), where (X, d) is a nonempty metric space and µ is a Borel measure on X. For every x ∈ X and r > 0, we write B(x, r) := {y ∈ X : d(x, y) < r} for the ball of radius r, and we also write V (x, r) := µ(B(x, r)) for the volume of this ball. The generalised half-space associated with X is the set X + := X × R + , equipped with the product topology and the product measure dµ(y) dt/t.

We say that (X, d, µ) is nondegenerate if

(3) 0 < V (x, r) < ∞ for all x ∈ X and r > 0.
This immediately implies that the measure space (X, µ) is σ-finite, as X may be written as an increasing sequence of balls

(4) X = n∈N B(x 0 , n)
for any point x 0 ∈ X. Nondegeneracy also implies that the metric space (X, d) is separable [START_REF] Björn | Nonlinear potential theory on metric spaces[END_REF]Proposition 1.6]. To rule out pathological behaviour (which is not particularly interesting from the viewpoint of tent spaces), we will always assume nondegeneracy. Generally we will need to make further geometric assumptions on (X, d, µ). In this article, the following two conditions will be used at various points. We say that (X, d, µ) is doubling if there exists a constant C ≥ 1 such that V (x, 2r) ≤ CV (x, r) for all (x, r) ∈ X + .

A consequence of the doubling condition is that there exists a minimal number n ≥ 0, called the doubling dimension of X, and a constant C ≥ 1 such that

V (x, R) ≤ C(R/r) n V (x, r) for all x ∈ X and 0 < r ≤ R < ∞.
For n > 0, we say that (X, d, µ) is AD-regular of dimension n if there exists a constant C ≥ 1 such that (5)

C -1 r n ≤ V (x, r) ≤ Cr n
for all x ∈ X and all r < diam(X). One can show that AD-regularity (of some dimension) implies doubling. Note that if X is unbounded and AD-regular of dimension n, then (5) holds for all x ∈ X and all r > 0.

1.2. Unweighted tent spaces. Throughout this section we suppose that (X, d, µ) is a nondegenerate metric measure space. We will not assume any further geometric conditions on X without explicit mention. All of the results here are known, at least in some form. We provide statements for ease of reference and some proofs for completeness.

For x ∈ X we define the cone with vertex x by Γ(x) := {(y, t) ∈ X + : y ∈ B(x, t)}, and for each ball B ⊂ X we define the tent with base B by

T (B) := X + \ x / ∈B Γ(x) .
Equivalently, T (B) is the set of points (y, t) ∈ X + such that B(y, t) ⊂ B. From this characterisation it is clear that if (y, t) ∈ T (B), then t ≤ r B , where we define r B := sup{r > 0 : B(y, r) ⊂ B for some y ∈ X}.

Note that it is possible to have r B(y,t) > t.

Fix q ∈ (0, ∞) and α ∈ R. For f ∈ L 0 (X + ), define functions A q f and C q α f on X by ( 6)

A q f (x) := ˆˆΓ(x) |f (y, t)| q dµ(y) V (y, t) dt t 1/q and (7) C q α f (x) := sup B∋x 1 µ(B) α 1 µ(B) ˆˆT(B)
|f (y, t)| q dµ(y) dt t

1/q for all x ∈ X, where the supremum in [START_REF] Blasco | Interpolation between vector-valued Hardy spaces[END_REF] is taken over all balls B ⊂ X containing x.

We abbreviate C q := C q 0 . Note that the integrals above are always defined (though possibly infinite) as the integrands are non-negative, and so we need not assume any local q-integrability of f . We also define (8)

A ∞ f (x) := ess sup (y,t)∈Γ(x) |f (y, t)| and C ∞ α f (x) := sup B∋x 1 µ(B) 1+α ess sup (y,t)∈T (B) |f (y, t)|.
Lemma 1.1. Suppose that q ∈ (0, ∞], α ∈ R, and f ∈ L 0 (X + ). Then the functions A q f and C q α f are lower semicontinuous. Proof. For q = ∞ see [START_REF] Amenta | Tent spaces over metric measure spaces under doubling and related assumptions[END_REF]Lemmas A.6 and A.7]. It remains only to show that A ∞ f and C ∞ α f are lower semicontinuous for f ∈ L 0 (X + ). For each s > 0 write

Γ(x) + s := {(y, t) ∈ X + : (y, t -s) ∈ Γ(x)} = {(y, t) ∈ X + : y ∈ B(x, t -s)}.
Geometrically Γ(x) + s is a 'vertically translated' cone, and Γ(x) + s ⊃ Γ(x) + r for all r < s. The triangle inequality implies that

Γ(x) + s ⊂ Γ(x ′ ) for all x ′ ∈ B(x, s). To show that A ∞ f is lower semicontinuous, suppose that x ∈ X and λ > 0 are such that (A ∞ f )(x) > λ. Then the set O := {(y, t) ∈ Γ(x) : |f (y, t)| > λ} has positive measure. We have O = ∞ n=1 O ∩ (Γ(x) + n -1 ).
Since the sequence of sets O∩(Γ(x)+n -1 ) is increasing in n, and since O has positive measure, we find that there exists n ∈ N such that O ∩ (Γ(x) + n -1 ) has positive measure. Thus for all x ′ ∈ B(x, n -1 ),

{(y, t) ∈ Γ(x ′ ) : |f (y, t)| > λ} ⊃ O ∩ (Γ(x) + n -1 )
has positive measure, and so

(A ∞ f )(x ′ ) > λ. Therefore A ∞ f is lower semicontinu- ous. The argument for C ∞ α is simpler. We have (C ∞ α f )(x) > λ if and only if there exists a ball B ∋ x such that 1 µ(B) 1+α ess sup (y,t)∈T (B) |f (y, t)| > λ. This immediately yields (C ∞ α f )(x ′ ) > λ for all x ′ ∈ B, and so C ∞ α f is lower semicon- tinuous.
Definition 1.2. For p ∈ (0, ∞) and q ∈ (0, ∞], the tent space T p,q (X) is the set

T p,q (X) := {f ∈ L 0 (X + ) : A q f ∈ L p (X)} equipped with the quasi-norm ||f || T p,q (X) := ||A q f || L p (X) .
We define T ∞,q (X) by

T ∞,q (X) := {f ∈ L 0 (X + ) : C q f ∈ L ∞ (X)}
equipped with the corresponding quasi-norm. We define T ∞,∞ (X) := L ∞ (X + ) with equal norms.

For the sake of notational clarity, we will write T p,q rather than T p,q (X) unless we wish to emphasise a particular choice of X. Although we will always refer to tent space 'quasi-norms', these are norms when p, q ≥ 1.

Remark 1.3. Our definition of A ∞ f gives a function which is less than or equal to the corresponding function defined by Coifman, Meyer, and Stein [START_REF]Some new function spaces and their applications to harmonic analysis[END_REF], which uses suprema instead of essential suprema. We also do not impose any continuity conditions in our definition of T p,∞ . Therefore our space T p,∞ (R n ) is strictly larger than the Coifman-Meyer-Stein version.

By a cylinder we mean a subset C ⊂ X + of the form C = B(x, r) × (a, b) for some (x, r) ∈ X + and 0 < a < b < ∞. We say that a function f ∈ L 0 (X + ) is cylindrically supported if it is essentially supported in a cylinder. In general cylinders may not be precompact, and so the notion of cylindrical support is more general than that of compact support. For all p, q ∈ (0, ∞] we define T p,q;c := {f ∈ T p,q : f is cylindrically supported}.

and

L p c (X + ) := {f ∈ L p (X + ) : f is cylindrically supported}. A straightforward application of the Fubini-Tonelli theorem shows that for all q ∈ (0, ∞) and for all f ∈ L 0 (X + ), ||f || T q,q = ||f || L q (X + ) , and so T q,q = L q (X + ). When q = ∞ this is true by definition.

Proposition 1.4. For all p, q ∈ (0, ∞), the subspace T p,q;c ⊂ T p,q is dense in T p,q . Furthermore, if X is doubling, then for all p, q ∈ (0, ∞], T p,q is complete, and when p, q = ∞, L q c (X + ) is densely contained in T p,q . Proof. The second statement has already been proven in [1, Proposition 3.5],1 so we need only prove the first statement. Suppose f ∈ T p,q and fix a point x 0 ∈ X. For each k ∈ N, define

C k := B(x 0 , k) × (k -1 , k) and f k := 1 C k f.
Then each f k is cylindrically supported. We have

lim k→∞ ||f -f k || p T p,q = lim k→∞ ˆX A q (1 C c k f )(x) p dµ(x) = ˆX lim k→∞ A q (1 C c k f )(x) p dµ(x) = ˆX lim k→∞ ˆˆΓ(x) |(1 C c k f )(y, t)| q dµ(y) V (y, t) dt t p/q dµ(x) = ˆX ˆˆΓ(x) lim k→∞ |(1 C c k f )(y, t)| q dµ(y) V (y, t) dt t p/q dµ(x) = 0.
All interchanges of limits and integrals follow from monotone convergence. Hence we have f = lim k→∞ f k , which completes the proof.

Recall the following duality from [1, Proposition 3.10].

Proposition 1.5. Suppose that X is doubling, p ∈ [1, ∞), and q ∈ (1, ∞). Then the L 2 (X + ) inner product

(9) f, g := ˆˆX + f (x, t)g(x, t) dµ(x) dt t
identifies the dual of T p,q with T p ′ ,q ′ . Suppose that p ∈ (0, 1], q ∈ [p, ∞], and B ⊂ X is a ball. We say that a function a ∈ L 0 (X + ) is a T p,q atom (associated with B) if a is essentially supported in T (B) and if the size estimate ||a|| T q,q ≤ µ(B) δp,q holds (recall that δ p,q := q -1 -p -1 ). A short argument shows that if a is a T p,q -atom, then ||a|| T p,q ≤ 1.

Theorem 1.6 (Atomic decomposition). Suppose that X is doubling. Let p ∈ (0, 1] and q ∈ [p, ∞]. Then a function f ∈ L 0 (X + ) is in T p,q if and only if there exists a sequence (a k ) k∈N of T p,q -atoms and a sequence (λ k ) k∈N ∈ ℓ p (N) such that

(10) f = k∈N λ k a k
with convergence in T p,q . Furthermore, we have

||f || T p,q ≃ inf ||λ k || ℓ p (N) ,
where the infimum is taken over all decompositions of the form [START_REF] Coifman | Analyse harmonique non-commutative sur certains espaces homogenes[END_REF]. This is proven by Russ when q = 2 [START_REF] Russ | The atomic decomposition for tent spaces on spaces of homogeneous type, CMA/AMSI research symposium "Asymptotic geometric analysis, harmonic analysis and related topics[END_REF], and the same proof works for general q ∈ [p, ∞). For q = ∞ we need to combine the original argument of Coifman, Meyer, and Stein [9, Proposition 2] with that of Russ. We defer this to Section 3.2. 1.3. Weighted tent spaces: definitions, duality, and atoms. We continue to suppose that (X, d, µ) is a nondegenerate metric measure space, and again we make no further assumptions without explicit mention.

For each s ∈ R, we can define an operator V s on L 0 (X + ) by

(V s f )(x, t) := V (x, t) s f (x, t)
for all (x, t) ∈ X + . Note that for r, s ∈ R the equality V r V s = V r+s holds, and also that V 0 is the identity operator. Using these operators we define modified tent spaces, which we call weighted tent spaces, as follows.

Definition 1.7. For p ∈ (0, ∞), q ∈ (0, ∞], and s ∈ R, the weighted tent space T p,q s is the set T p,q s := {f ∈ L 0 (X + ) : V -s f ∈ T p,q } equipped with the quasi-norm

||f || T p,q s := V -s f T p,q .
We also define T ∞,∞ s in this way. For q = ∞, and with an additional parameter α ∈ R, we define T ∞,q s;α by the quasi-norm ||f || T ∞,q s;α :

= C q α (V -s f ) L ∞ (X) .
Note that T ∞,q 0;0 = T ∞,q . We write T ∞,q s := T ∞,q s;0 . Remark 1.8. The weighted tent space quasi-norms of Hofmann, Mayboroda, and McIntosh [14, §8.3] (with p = ∞) and Huang [START_REF] Huang | Weighted tent spaces with Whitney averages: Factorization, interpolation and duality[END_REF] (including p = ∞ with α = 0) are given by [START_REF] Gilbert | Interpolation between weighted L p -spaces[END_REF] ||f || T p,q s (R n ) := (y, t) → t -s f (y, t) T p,q (R n ) , which are equivalent to those of our spaces T p,q s/n (R n ). In general, when X is unbounded and AD-regular of dimension n, the quasi-norm in [START_REF] Gilbert | Interpolation between weighted L p -spaces[END_REF] (with X replacing R n ) is equivalent to that of our T p,q s/n . We have chosen the convention of weighting with ball volumes, rather than with the variable t, because this leads to more geometrically intrinsic function spaces and supports embedding theorems under weaker assumptions.

For all r, s ∈ R, the operator V r is an isometry from T p,q s to T p,q s+r . The operator V -r is also an isometry, now from T p,q s+r to T p,q s , and so for fixed p and q the weighted tent spaces T p,q s are isometrically isomorphic for all s ∈ R. Thus by Proposition 1.4, when X is doubling, the spaces T p,q s are all complete. Recall the L 2 (X + ) inner product [START_REF]Some new function spaces and their applications to harmonic analysis[END_REF], which induces a duality pairing between T p,q and T p ′ ,q ′ for appropriate p and q when X is doubling. For all s ∈ R and all f, g ∈ L 2 (X + ) we have the equality [START_REF] Harboure | A vector-valued approach to tent spaces[END_REF] f, g = V -s f, V s g , which yields the following duality result.

Proposition 1.9. Suppose that X is doubling, p ∈ [1, ∞), q ∈ (1, ∞), and s ∈ R.

Then the L 2 (X + ) inner product (9) identifies the dual of T p,q s with T p ′ ,q ′ -s .

Proof. If f ∈ T p,q s and g ∈ T p ′ ,q ′ -s , then we have V -s f ∈ T p,q and V s g ∈ T p ′ ,q ′ , so by Proposition 1.5 and ( 12) we have

| f, g | V -s f T p,q ||V s g|| T p ′ ,q ′ = ||f || T p,q s ||g|| T p ′ ,q ′ -s
.

Conversely, if ϕ ∈ (T p,q s ) ′ , then the map f → ϕ(V s f ) determines a bounded linear functional on T p,q with norm dominated by ||ϕ||. Hence by Proposition 1.5 there exists a function g ∈ T p ′ ,q ′ with ||g|| T p ′ ,q ′ ||ϕ|| such that

ϕ(f ) = ϕ(V s (V -s f )) = V -s f, g = f, V -s g for all f ∈ T p,q s . Since V -s g T p ′ ,q ′ -s = ||g|| T p ′ ,q ′ ||ϕ|| , we are done.
There is also a duality result for p < 1 which incorporates the spaces T ∞,q s;α with α > 0. Before we can prove it, we need to discuss atomic decompositions.

Suppose that p ∈ (0, 1], q ∈ [p, ∞], s ∈ R, and B ⊂ X is a ball. We say that a function a ∈ L 0 (X + ) is a T p,q s -atom (associated with B) if V -s a is a T p,q -atom. This is equivalent to demanding that a is essentially supported in T (B) and that

||a|| T q,q s ≤ µ(B) δp,q .
The atomic decomposition theorem for unweighted tent spaces (Theorem 1.6) immediately implies its weighted counterpart.

Proposition 1.10 (Atomic decomposition for weighted tent spaces). Suppose that

X is doubling. Let p ∈ (0, 1], q ∈ [p, ∞], and s ∈ R. Then a function f ∈ L 0 (X + ) is in T p,q s
if and only if there exists a sequence (a k ) k∈N of T p,q s -atoms and a sequence

(λ k ) k∈N ∈ ℓ p (N) such that (13) f = k∈N λ k a k
with convergence in T p,q s . Furthermore, we have

||f || T p,q s ≃ inf ||λ k || ℓ p (N) ,
where the infimum is taken over all decompositions of the form [START_REF] Hofmann | Hardy and BMO spaces associated to divergence form elliptic operators[END_REF].

Using this, we can prove the following duality result for p < 1.

Theorem 1.11. Suppose that X is doubling, p ∈ (0, 1), q ∈ [1, ∞), and s ∈ R.

Then the L 2 (X + ) inner product (9) identifies the dual of T p,q s with T ∞,q ′ -s;δ 1,p .

Proof. First suppose that a is a T p,q s -atom associated with a ball B ⊂ X, and that g ∈ T ∞,q ′ -s,δ 1,p . Then we have

| a, g | ≤ ˆˆT(B) |V -s a(y, t)||V s g(y, t)| dµ(y) dt t ≤ ||a|| T q,q s µ(B) 1/q ′ µ(B) δ 1,p ||g|| T ∞,q ′ -s,δ 1,p ≤ µ(B) δp,q+δ q,1 +δ 1,p ||g|| T ∞,q ′ -s,δ 1,p = ||g|| T ∞,q ′ -s,δ 1,p .
For general f ∈ T p,q s we write f as a sum of T p,q s -atoms as in ( 13) and get

| f, g | ≤ ||g|| T ∞,q ′ -s,δ 1,p ||λ|| ℓ 1 ≤ ||g|| T ∞,q ′ -s,δ 1,p ||λ|| ℓ p using that p < 1.
Taking the infimum over all atomic decompositions completes the argument.

Conversely, suppose that ϕ ∈ (T p,q s ) ′ . Exactly as in the classical duality proof (see [1, Proof of Proposition 3.10]), using the doubling assumption, there exists a function g ∈ L q ′ loc (X + ) such that

ϕ(f ) = f, g
for all f ∈ T p,q;c s . To show that g is in T ∞,q ′ -s,δ 1,p , we estimate ||V s g|| L q ′ (T (B)) for each ball B ⊂ X by duality:

||V s g|| L q ′ (T (B)) = sup f ∈L q (T (B)) | f, V s g | ||f || -1 L q (T (B)) = sup f ∈L q c (T (B)) | V s f, g | ||f || -1 L q (T (B)) .
Hölder's inequality implies that

||V s f || T p,q s ≤ µ(B) δq,p ||f || L q (T (B))
when f is essentially supported in T (B), so we have

||V s g|| L q ′ (T (B)) ≤ µ(B) δq,p ||ϕ|| (T p,q s ) ′ , and therefore ||g|| T ∞,q ′ -s,δ 1,p = sup B⊂X µ(B) δ p,1 -(1/q ′ ) ||V s g|| L q ′ (T (B)) ≤ ||ϕ|| (T p,q s ) ′ sup B⊂X µ(B) δ p,1 +δ 1,q +δq,p
= ||ϕ|| (T p,q s ) ′ , which completes the proof.

Remark 1.12. Note that q = 1 is included here, and excluded in the other duality results of this article. Generally the spaces T p,q with p ≤ q are easier to handle than those with p > q.

We end this section by detailing a technique, usually referred to as 'convex reduction', which is very useful in relating tent spaces to each other. Suppose f ∈ L 0 (X + ) and M > 0. We define a function f M ∈ L 0 (X + ) by

(f M )(x, t) := |f (x, t)| M
for all (x, t) ∈ X + . For all q ∈ (0, ∞] and s ∈ R we then have

A q (V -s f M ) = A M q (V -s/M f ) M ,
and for α ∈ R we also have

C q α (V -s f M ) = C M q α/M (V -s/M f ) M . Therefore, for p ∈ (0, ∞) we have f M T p,q s = A M q (V -s/M f ) M L p (X) = A M q (V -s/M f ) M L M p (X) = f | T M p,M q s/M M ,
and likewise for p = ∞ and q < ∞ we have

f M T ∞,q s,α = f | T ∞,M q s/M,α/M M
The case p = q = ∞ behaves in the same way:

f M T ∞,∞ s = (V -s/M f ) M L ∞ (X + ) = ||f || M T ∞,∞ s/M .
These equalities often allow us to deduce properties of T p,q s from properties of T M p,M q s/M , and vice versa. We will use them frequently.

Interpolation and embeddings

As always, we assume that (X, d, µ) is a nondegenerate metric measure space. We will freely use notation and terminology regarding interpolation theory; the uninitiated reader may refer to Bergh and Löfström [START_REF] Bergh | Interpolation spaces, Grundlehren der mathematischen Wissenschaften[END_REF].

2.1. Complex interpolation. In this section we will make the following identification of the complex interpolants of weighted tent spaces in the Banach range of exponents.

Theorem 2.1. Suppose that X is doubling, p 0 , p 1 ∈ [1, ∞] (not both ∞), q 0 , q 1 ∈ (1, ∞), s 0 , s 1 ∈ R,
and θ ∈ (0, 1). Then we have the identification

[T p 0 ,q 0 s 0 , T p 1 ,q 1 s 1 ] θ = T p θ ,q θ s θ where p -1 θ = (1 -θ)p -1 0 + θp -1 1 , q -1 θ = (1 -θ)q -1 0 + θq -1 1
, and s θ = (1 -θ)s 0 + θs 1 . Remark 2.2. In the case where X = R n with the Euclidean distance and Lebesgue measure, this result (with p 0 , p 1 < 1 permitted) is due to Hofmann, Mayboroda, and McIntosh [START_REF] Hofmann | Second order elliptic operators with complex bounded measurable coefficients in L p , Sobolev, and Hardy spaces[END_REF]Lemma 8.23]. A more general result, still with X = R n , is proven by Huang [15, Theorem 4.3] with q 0 , q 1 = ∞ also permitted, and with Whitney averages incorporated. Both of these results are proven by means of factorisation theorems for weighted tent spaces (with Whitney averages in the second case), and by invoking an extension of Calderón's product formula to quasi-Banach spaces due to Kalton and Mitrea [START_REF] Kalton | Stability results on interpolation scales of quasi-Banach spaces and applications[END_REF]Theorem 3.4]. We have chosen to stay in the Banach range with 1 < q 0 , q 1 < ∞ for now, as establishing a general factorisation result would take us too far afield.

Note that if p 0 = ∞ (say) then we are implicitly considering T ∞,q 0 s 0 ;α with α = 0; interpolation of spaces with α = 0 is not covered by this theorem. This is because the method of proof uses duality, and to realise T ∞,q 0 s 0 ;α with α = 0 as a dual space we would need to deal with complex interpolation of quasi-Banach spaces, which adds difficulties that we have chosen to avoid.

Before moving on to the proof of Theorem 2.1, we must fix some notation. For q ∈ (1, ∞) and s ∈ R, write

(14) L q s (X + ) := L q (X + , V -qs-1 ) := L q X + , V -qs (y, t) dµ(y) V (y, t) dt t
(this notation is consistent with viewing the function V -qs-1 as a weight on the product measure dµ dt/t).

An important observation, originating from Harboure, Torrea, and Viviani [START_REF] Harboure | A vector-valued approach to tent spaces[END_REF], is that for all p ∈ [1, ∞), q ∈ (1, ∞) and s ∈ R, one can write

||f || T p,q s = Hf | L p (X : L q s (X + )) for f ∈ L 0 (X + ), where
Hf (x) = 1 Γ(x) f. Hence H is an isometry from T p,q s to L p (X : L q s (X + )). Because of the restriction on q, the theory of Lebesgue spaces (more precisely, Bochner spaces) with values in reflexive Banach spaces is then available to us. This proof follows previous arguments of the author [START_REF] Amenta | Tent spaces over metric measure spaces under doubling and related assumptions[END_REF], which are based on the ideas of Harboure, Torrea, and Viviani [START_REF] Harboure | A vector-valued approach to tent spaces[END_REF] and of Bernal [START_REF] Bernal | Some results on complex interpolation of T p q spaces, Interpolation spaces and related topics (Ramat-Gan)[END_REF], with only small modifications to incorporate additional parameters. We include it to show where these modifications occur: in the use of duality, and in the convex reduction.

Proof of Theorem 2.1. First we will prove the result for p 0 , p 1 ∈ (1, ∞). Since H is an isometry from T p j ,q j s j to L p j (X : L q j s j (X + )) for j = 0, 1, the interpolation property implies that H is bounded (with norm ≤ 1 due to exactness of the complex interpolation functor)

[T p 0 ,q 0 s 0 , T p 1 ,q 1 s 1 ] θ → L p θ X : [L q 0 s 0 (X + ), L q 1 s 1 (X + )] θ .
Here we have used the standard identification of complex interpolants of Banachvalued Lebesgue spaces [4, Theorem 5.1.2]. The standard identification of complex interpolants of weighted Lebesgue spaces [START_REF] Bergh | Interpolation spaces, Grundlehren der mathematischen Wissenschaften[END_REF]Theorem 5.5.3] gives

[L q 0 s 0 (X + ), L q 1 s 1 (X + )] θ = L q θ s θ (X +
), and we conclude that

||f || T p θ ,q θ s θ = Hf | L p θ (X : L q θ s θ (X + )) ≤ f | [T p 0 ,q 0 s 0 , T p 1 ,q 1 s 1 ] θ for all f ∈ [T p 0 ,q 0 s 0 , T p 1 ,q 1 s 1 ] θ . Therefore (15) [T p 0 ,q 0 s 0 , T p 1 ,q 1 s 1 ] θ ⊂ T p θ ,q θ s θ .
To obtain the reverse inclusion, we use the duality theorem for complex interpolation [4, Theorem 4.5.1 and Corollary 4.5.2]. Since X is doubling and by our restrictions on p and q, at least one of the spaces T p 0 ,q 0 s 0 and T p 1 ,q 1 s 1 is reflexive (by Proposition 1.9) and their intersection is dense in both spaces (as it contains the dense subspace L max(q 0 ,q 1 ) c (X + ) by Proposition 1.4). Therefore the assumptions of the duality theorem for complex interpolation are satisfied, and we have

T p θ ,q θ s θ = (T p ′ θ ,q ′ θ -s θ ) ′ ⊂ [T p ′ 0 ,q ′ 0 -s 0 , T p ′ 1 ,q ′ 1 -s 1 ] ′ θ = [T p 0 ,q 0 s 0 , T p 1 ,q 1 s 1 ] θ
where the first two lines follow from Proposition (1.9) and ( 15), and the third line uses the duality theorem for complex interpolation combined with Proposition 1.9.

We can extend this result to p 0 , p 1 ∈ [1, ∞] using the technique of [1, Proposition 3.18]. The argument is essentially identical, so we will not include the details here.

2.2.

Real interpolation: the reflexive range. In order to discuss real interpolation of weighted tent spaces, we need to introduce a new scale of function spaces, which we denote by Z p,q s = Z p,q s (X). 2 Definition 2.3. For c 0 ∈ (0, ∞), c 1 ∈ (1, ∞), and (x, t) ∈ X + , we define the Whitney region 2 We use this notation because almost every other reasonable letter seems to be taken.

Ω c 0 ,c 1 (x, t) := B(x, c 0 t) × (c -1 1 t, c 1 t) ⊂ X + ,
and for q ∈ (0, ∞), f ∈ L 0 (X + ), and (x, t) ∈ X + we define the L q -Whitney average

(W q c 0 ,c 1 f )(x, t) := ˆˆΩc 0 ,c 1 (x,t) |f (ξ, τ )| q dµ(ξ) dτ 1/q . For p, q ∈ (0, ∞), s ∈ R, c 0 ∈ (0, ∞), c 1 ∈ (1, ∞)
, and f ∈ L 0 (X + ), we then define the quasi-norm ||f || Z p,q s (X;c 0 ,c 1 ) := W q c 0 ,c 1 (V -s f ) L p (X + ) . and the Z-space Z p,q s (X; c 0 , c 1 ) := {f ∈ L 0 (X + ) : ||f || Z p,q s (X;c 0 ,c 1 ) < ∞}. In this section we will prove the following theorem, which identifies real interpolants of weighted tent spaces in the reflexive range. We will extend this to the full range of exponents in the Euclidean case in the next section.

Theorem 2.4. Suppose that X is AD-regular and unbounded, p 0 , p 1 , q ∈ (1, ∞), s 0 = s 1 ∈ R, and θ ∈ (0, 1). Then for any c 0 ∈ (0, ∞) and c 1 ∈ (1, ∞) we have the identification [START_REF] Jawerth | Some observations on Besov and Lizorkin-Triebel spaces[END_REF] (T p 0 ,q s 0 , T p 1 ,q s 1 ) θ,p θ = Z p θ ,q s θ (X; c 0 , c 1 ) with equivalent norms, where

p -1 θ = (1 -θ)p -1 0 + θp -1 1 and s θ = (1 -θ)s 0 + θs 1 .
As a corollary, in the case when X is AD-regular and unbounded, and when p, q > 1, the spaces Z p,q s (X; c 0 , c 1 ) are independent of the parameters (c 0 , c 1 ) with equivalent norms, and we can denote them all simply by Z p,q s . 3 We remark that most of the proof does not require AD-regularity, but in its absence we obtain identifications of the real interpolants which are less convenient.

The proof relies on the following identification of real interpolants of weighted L q spaces, with fixed q and distinct weights, due to Gilbert [START_REF] Gilbert | Interpolation between weighted L p -spaces[END_REF]Theorem 3.7]. The cases p ≤ 1 and q < 1 are not considered there, but the proof still works without any modifications in these cases. Note that the original statement of this theorem contains a sign error in the expression corresponding to [START_REF] Kalton | Quasi-Banach spaces[END_REF].

Theorem 2.5 (Gilbert). Suppose (M, µ) is a σ-finite measure space and let w be a weight on (M, µ). Let p, q ∈ (0, ∞) and θ ∈ (0, 1). For all r ∈ (1, ∞), and for f ∈ L 0 (M), the expressions

(17) r -kθ 1 x:w(x)∈(r -k ,r -k+1 ] f L q (M ) k∈Z ℓ p (Z) (18) s 1-θ 1 x:w(x)≤1/s f L q (M,w q ) L p (R + ,ds/s)
and

(19) s -θ 1 x:w(x)>1/s f L q (M ) L p (R + ,ds/s)
define equivalent norms on the real interpolation space (L q (M), L q (M, w q )) θ,p .

3 One can prove independence of the parameters (c 0 , c 1 ) directly when X is doubling, but proving this here would take us even further off course.

The first step in the proof of Theorem 2.4 is a preliminary identification of the real interpolation norm. Proposition 2.6. Let all numerical parameters be as in the statement of Theorem 2.4. Then for all f ∈ L 0 (X + ) we have the equivalence [START_REF] Pisier | The K t -functional for the interpolation couple L 1 (A 0 ), L ∞ (A 1 )[END_REF] f | (T p 0 ,q s 0 , T p 1 ,q s 1 ) θ,

p θ ≃ x → 1 Γ(x) f | (L q s 0 (X + ), L q s 1 (X + )) θ,p θ L p θ (X) .
Proof. We use the notation of the previous section. We have already noted that the map H : T p,q s → L p (X : L q s (X + )) with Hf (x) = 1 Γ(x) f is an isometry. Furthermore, as shown in [START_REF] Amenta | Tent spaces over metric measure spaces under doubling and related assumptions[END_REF] (see the discussion preceding Proposition 3.12 there), H(T p,q s ) is complemented in L p (X : L q s (X + )), and there is a common projection onto these spaces. Therefore we have (by [22, Theorem 1.17.1.1] for example) f | (T p 0 ,q s 0 , T p 1 ,q s 1 ) θ,p θ ≃ Hf | (L p 0 (X : L q s 0 (X + )), L p 1 (X : L q s 1 (X + ))) θ,p θ . The Lions-Peetre result on real interpolation of Banach-valued Lebesgue spaces (see for example [START_REF] Pisier | The K t -functional for the interpolation couple L 1 (A 0 ), L ∞ (A 1 )[END_REF]Remark 7]) then implies that f | (T p 0 ,q s 0 , T p 1 ,q s 1 ) θ,p θ ≃ Hf | (L p θ (X : (L q s 0 (X + ), L q s 1 (X + )) θ,p θ ) . Since Hf (x) = 1 Γ(x) f , this proves [START_REF] Pisier | The K t -functional for the interpolation couple L 1 (A 0 ), L ∞ (A 1 )[END_REF].

Having proven Proposition 2.6, we can use Theorem 2.5 to provide some useful characterisations of the real interpolation norm. For f ∈ L 0 (X + ) and a, b ∈ [0, ∞], we define the truncation

f a,b := 1 X×(a,b) f.
Note that in this theorem we allow for p 0 , p 1 ≤ 1; we will use this range of exponents in the next section.

Theorem 2.7. Suppose p 0 , p 1 , q ∈ (0, ∞), s 0 = s 1 ∈ R, and θ ∈ (0, 1), and suppose that X is AD-regular of dimension n and unbounded. Let r ∈ (1, ∞). Then for f ∈ L 0 (X + ) we have norm equivalences

x → 1 Γ(x) f | (L q s 0 (X + ), L q s 1 (X + )) θ,p θ L p θ (X) ≃ τ n(s 1 -s 0 )(1-θ) ||f τ,∞ || T p θ ,q s 1 L p θ (R + ,dτ /τ ) (21)
≃ τ -n(s 1 -s 0 )θ ||f 0,τ || T p θ ,q s 0 L p θ (R + ,dτ /τ ) (22) ≃ r (r -nkθ(s 1 -s 0 ) f r -k ,r -k+1 T p θ ,q s 0 ) k∈Z ℓ p θ (Z) . ( 23
)
Proof. First assume that s 1 > s 0 . Let µ q s 0 be the measure on X + given by dµ q s 0 (y, t) := t -qs 0 n dµ(y) dt V (y, t)t .

Since X is AD-regular of dimension n and unbounded, we have that ||f || L q (µ q s 0 ) ≃ ||f || L q s 0 (X + ) . Also define the weight w(y, t) := t -(s 1 -s 0 )n , so that w q µ q s 0 = µ q s 1 . We will obtain the norm equivalence (23). For 1 < r < ∞ and k ∈ Z, we have r -k < w(y, t) ≤ r -k+1 if and only if t ∈ [r (k-1)/n(s 1 -s 0 ) , r k/n(s 1 -s 0 ) ) (here we use s 1 > s 0 ). Using the characterisation (17) of Theorem 2.5, and replacing r with r n(s 1 -s 0 ) , for f ∈ L 0 (X + ) we have

x → 1 Γ(x) f | (L q s 0 (X + ), L q s 1 (X + )) θ,p θ L p θ (X) ≃ ˆX 1 Γ(x) f
p θ (L q (µ q s 0 ),L q (w q µ q s 0 )) θ,p θ dµ(x)

1/p θ ≃ ˆX k∈Z r -n(s 1 -s 0 )kθp θ 1 Γ(x) f r k-1 ,r k p θ L q (µ q s 0 ) dµ(x) 1/p θ ≃ k∈Z r -n(s 1 -s 0 )kθp θ ˆX A q (V -s 0 f r k-1 ,r k )(x) p θ dµ(x) 1/p θ = (r -n(s 1 -s 0 )kθ f r k-1 ,r k T p θ ,q s 0 ) k∈Z ℓ p θ (Z)
.

This proves the norm equivalence (23) for all f ∈ L 0 (X + ) when s 1 > s 0 . If s 1 < s 0 , one simply uses the identification (L q s 0 (X + ), L q s 1 (X + )) θ,p θ = (L q s 1 (X + ), L q s 0 (X + )) 1-θ,p θ [4, Theorem 3.4.1(a)] to reduce the problem to the case where s 0 < s 1 .

The equivalences ( 21) and ( 22) follow from the characterisations ( 18) and ( 19) of Theorem 2.5 in the same way, with integrals replacing sums throughout. We omit the details here.

Finally we can prove the main theorem: the identification of the real interpolants of weighted tent spaces as Z-spaces.

Proof of Theorem 2.4. Suppose f ∈ L 0 (X + ). Using the characterisation (23) in Theorem 2.7 with r = c 1 > 1, and using aperture c 0 /c 1 for the tent space (making use of the change of aperture theorem [1, Proposition 3.21]), we have

f | (T p 0 ,q s 0 , T p 1 ,q s 1 ) θ,p θ p θ ≃ k∈Z c -n(s 1 -s 0 )kθp θ 1 ˆX ˆck 1 c k-1 1 ˆB(x,c 0 t/c 1 ) |t -ns 0 f (y, t)| q dµ(y) V (y, t) dt t p θ /q dµ(x) ≃ ˆX k∈Z c -n(s 1 -s 0 )kθp θ 1 • ˆck 1 c k-1 1 ˆck 1 c k-1 1 ˆB(x,c 0 t/c 1 ) |t -ns 0 f (y, t)| q dµ(y) V (y, t) dt t p θ /q dr r dµ(x) ˆX k∈Z c -n(s 1 -s 0 )kθp θ 1 ˆck 1 c k-1 1 ˆˆΩc 0 ,c 1 (x,r) |r -ns 0 f (y, t)| q dµ(y) dt p θ /q dr r dµ(x) ≃ ˆX ˆ∞ 0 r -n(s 1 -s 0 )θp θ ˆˆΩc 0 ,c 1 (x,r) |r -ns 0 f | q p θ /q dr r dµ(x) = ˆˆX + ˆˆΩc 0 ,c 1 (x,r) |r -ns θ f | q p θ /q dµ(x) dr r ≃ ||f || p θ Z p θ ,q s θ (X;c 0 ,c 1 ) , using that B(x, c 0 t/c 1 ) × (c k-1 1 , c k 1 ) ⊂ Ω c 0 ,c 1 (x, r) whenever r ∈ (c k-1 1 , c k 1 ).
To prove the reverse estimate we use the same argument, this time using that for r, t ∈ (2 k-1 , 2 k ) we have Ω c 0 ,c 1 (x, t) ⊂ B(x, 2c 0 t) × (c -1 1 2 k-1 , c 1 2 k ). Using aperture 2c 0 for the tent space, we can then conclude that

||f || p θ Z p θ ,q s θ (X;c 0 ,c 1 ) ≃ ˆX k∈Z 2 -n(s 1 -s 0 )kθp θ ˆ2k 2 k-1 ˆˆΩc 0 ,c 1 (x,r) |r -ns 0 f | q p θ /q dr r dµ(x) ˆX k∈Z 2 -n(s 1 -s 0 )kθp θ ˆ2k 2 k-1 ˆc1 2 k c -1 1 2 k-1 ˆB(x,2c 0 t) |r -ns 0 f (y, t)| q dµ(y) V (y, t) dt t p θ /q dr r dµ(x) ≃ ˆX k∈Z 2 -n(s 1 -s 0 )kθp θ ˆc1 2 k c -1 1 2 k-1 ˆc1 2 k c -1 1 2 k-1 ˆB(x,2c 0 t) |r -ns 0 f (y, t)| q dµ(y) V (y, t) dt t p θ /q dr r dµ(x) ≃ f | (T p 0 ,q s 0 , T p 1 ,q s 1 ) θ,p θ p θ .
This completes the proof of Theorem 2.4.

Remark 2.8. Note that this argument shows that

(r -nkθ(s 1 -s 0 ) f r -k ,r -k+1 T p θ ,q s 0 ) k∈Z ℓ p θ (Z) ≃ ||f || Z p θ ,q s θ (X;c 0 ,c 1 )
whenever X is AD-regular of dimension n and unbounded, for all p 0 , p 1 ∈ (0, ∞), c 0 ∈ (0, ∞), and c 1 ∈ (1, ∞). Therefore, since Theorem 2.7 also holds for this range of exponents, to establish the identification (24) for p 0 , p 1 ∈ (0, ∞) it suffices to extend Proposition 2.6 to p 0 , p 1 ∈ (0, ∞). We will do this in the next section in the Euclidean case.

2.3.

Real interpolation: the non-reflexive range. In this section we prove the following extension of Theorem 2. [START_REF] Bergh | Interpolation spaces, Grundlehren der mathematischen Wissenschaften[END_REF]. In what follows, we always consider R n as a metric measure space with the Euclidean distance and Lebesgue measure.

Theorem 2.9. Suppose that p 0 , p 1 ∈ (0, ∞), q ∈ [1, ∞), s 0 = s 1 ∈ R, and θ ∈ (0, 1).

Then for any c 0 ∈ (0, ∞) and c 1 ∈ (1, ∞) we have the identification

(24) (T p 0 ,q s 0 (R n ), T p 1 ,q s 1 (R n )) θ,p θ = Z p θ ,q s θ (R n ; c 0 , c 1 ) with equivalent quasi-norms, where p -1 θ = (1 -θ)p -1 0 + θp -1 1 and s θ = (1 -θ)s 0 + θs 1 .
The main difficulty here is that vector-valued Bochner space techniques are not available to us, as we would need to use quasi-Banach valued L p spaces with p < 1, and such a theory is not well-developed. Furthermore, although the weighted tent spaces T p,q s embed isometrically into L p (X : L q s (X + )) in this range of exponents, their image may not be complemented, and so we cannot easily identify interpolants of their images. 4 We must argue directly.

First we recall the so-called 'power theorem' [START_REF] Bergh | Interpolation spaces, Grundlehren der mathematischen Wissenschaften[END_REF]Theorem 3.11.6], which allows us to exploit the convexity relations between weighted tent spaces. If A is a quasi-Banach space with quasi-norm ||•|| and if ρ > 0, then ||•|| ρ is also a quasi-norm on A, and we denote the resulting quasi-Banach space by A ρ . Theorem 2.10 (Power theorem). Let (A 0 , A 1 ) be a compatible couple of quasi-Banach spaces. Let ρ 0 , ρ 1 ∈ (0, ∞), η ∈ (0, 1), and r ∈ (0, ∞], and define ρ := (1 -η)ρ 0 + ηρ 1 , θ := ηρ 1 /ρ, and σ := rρ. Then we have

((A 0 ) ρ 0 , (A 1 ) ρ 1 ) η,r = ((A 0 , A 1 ) θ,σ ) ρ
with equivalent quasi-norms.

Before proving Theorem 2.9 we must establish some technical lemmas. Recall that we previously defined the spaces L q s (X + ) in ( 14). Lemma 2.11. Suppose x ∈ X, α ∈ (0, ∞), and let all other numerical parameters be as in the statement of Theorem 2.9. Then for all cylindrically supported f ∈ L 0 (X + ) we have

K(α, 1 Γ(x) f ; L q s 0 (X + ), L q s 1 (X + )) = inf f =ϕ 0 +ϕ 1 A q (V -s 0 ϕ 0 )(x) + αA q (V -s 1 ϕ 1 )(x) (25)
and

K(α, 1 Γ(x) f ; L q s 0 (X + ) p 0 , L q s 1 (X + ) p 1 ) = inf f =ϕ 0 +ϕ 1 A q (V -s 0 ϕ 0 )(x) p 0 + αA q (V -s 1 ϕ 1 )(x) p 1 (26)
where the infima are taken over all decompositions f = ϕ 0 + ϕ 1 in L 0 (X + ) with ϕ 0 , ϕ 1 cylindrically supported.

Proof. We will only prove the equality (25), as the proof of ( 26) is essentially the same.

Given a decomposition f = ϕ 0 + ϕ 1 in L 0 (X + ), we have a corresponding decomposition 1 Γ(x) f = 1 Γ(x) ϕ 0 + 1 Γ(x) ϕ 1 , with 1 Γ(x) ϕ 0 L q s 0 (X + ) = A q (V -s 0 ϕ 0 )(x) and likewise for ϕ 1 . This shows that

K(α, 1 Γ(x) f ; L q s 0 (X + ), L q s 1 (X + )) ≤ inf f =ϕ 0 +ϕ 1 A q (V -s 0 ϕ 0 )(x) + αA q (V -s 1 ϕ 1 )(x) .
For the reverse inequality, suppose that 1 Γ(x) f = ϕ 0 + ϕ 1 in L 0 (X + ), and suppose f is essentially supported in a cylinder C. Multiplication by the characteristic function 1 Γ(x)∩C does not increase the quasi-norms of ϕ 0 and ϕ 1 in L q s 0 (X + ) and L q s 1 (X + ) respectively, so without loss of generality we can assume that ϕ 0 and ϕ 1 are cylindrically supported in Γ(x). Now let f = ψ 0 + ψ 1 be an arbitrary decomposition in L 0 (X + ), and define

ψ 0 := 1 Γ(x) ϕ 0 + 1 X + \Γ(x) ψ 0 , ψ 1 := 1 Γ(x) ϕ 1 + 1 X + \Γ(x) ψ 1 .
Then f = ψ 0 + ψ 1 in L 0 (X + ), and we have

A q (V -s 0 ψ 0 )(x) = A q (V -s 0 ϕ 0 )(x) = 1 Γ(x) ϕ 0 L q s 0 (X + )
and likewise for ψ 1 . The conclusion follows from the definition of the K-functional. Lemma 2.12. Suppose f ∈ L q c (X + ). Then A q f is continuous.

Proof. Let f be essentially supported in the cylinder C := B(c, r) × (κ 0 , κ 1 ). First, for all x ∈ X we estimate

A q f (x) ≤ ˆˆC |f (y, t)| q dµ(y) V (y, t) dt t 1/q ≤ inf y∈B V (y, κ 0 ) -1/q ||f || L q (X + ) ||f || L q (X + ) ,
using the estimate (40) from the proof of Lemma 3.1. For all x ∈ X we thus have

lim z→x |A q f (x) -A q f (z)| ≤ lim z→x ˆˆX + |1 Γ(x) -1 Γ(z) ||f (y, t)| q dµ(y) V (y, t) dt t 1/q = 0
by dominated convergence, since 1 Γ(x) -1 Γ(z) → 0 pointwise as z → x, and since

ˆˆX + |1 Γ(x) -1 Γ(z) ||f (y, t)| q dµ(y) V (y, t) dt t 1/q ||f || L q (X + ) .
Therefore A q f is continuous.

Having established these lemmas, we can prove the following (half-)extension of Proposition 2.6. Proposition 2.13. Let all numerical parameters be as in the statement of Theorem 2.9. Then for all f ∈ L q c (X + ) the function (27)

x → 1 Γ(x) f | (L q s 0 (X + ), L q s 1 (X + )) θ,p θ is measurable on X (using the discrete characterisation of the real interpolation quasi-norm), and we have

(28) f | (T p 0 ,q s 0 , T p 1 ,q s 1 ) θ,p θ x → 1 Γ(x) f | (L q s 0 (X + ), L q s 1 (X + )) θ,p θ L p θ (X) .
We denote the quantity on the right hand side of (20) by f | I p θ ,q s 0 ,s 1 ,θ . Proof. First we take care of measurability. Using Lemma 2.11, for x ∈ X we write

1 Γ(x) f | (L q s 0 (X + ), L q s 1 (X + )) θ,p θ p θ = k∈Z 2 -kp θ θ K 2 k , 1 Γ(x) f ; L q s 0 (X + ), L q s 1 (X + ) p θ = k∈Z 2 -kp θ θ inf f =ϕ 0 +ϕ 1 A q (V -s 0 ϕ 0 )(x) + 2 k A q (V -s 1 ϕ 1 )(x) p θ
where the infima are taken over all decompositions f = ϕ 0 + ϕ 1 in L 0 (X + ) with ϕ 0 ∈ L q s 0 (X + ) and ϕ 1 ∈ L q s 1 (X + ) cylindrically supported. By Lemma 2.12, we have that A q (V -s 0 ϕ 0 ) and A q (V -s 1 ϕ 1 ) are continuous. Hence for each k ∈ Z and for every such decomposition f = ϕ 0 + ϕ 1 the function x → A q (V -s 0 ϕ 0 )(x) + 2 k A q (V -s 1 ϕ 1 )(x) is continuous. The infimum of these functions is then upper semicontinuous, therefore measurable.

Next, before beginning the proof of the estimate (28), we apply the power theorem with A 0 = T p 0 ,q s 0 , A 1 = T p 1 ,q s 1 , ρ 0 = p 0 , ρ 1 = p 1 , and σ = p θ . Then we have ρ = p θ , η = θp θ /p 1 , r = 1, and the relation p θ = (1 -η)p 0 + ηp 1 is satisfied. We conclude that ((T p 0 ,q s 0 , T p 1 ,q s 1 ) θ,p θ ) p θ ≃ ((T p 0 ,q s 0 ) p 0 , (T p 1 ,q s 1 ) p 1 ) θp θ /p 1 ,1 . Thus it suffices for us to prove (29) f | ((T p 0 ,q s 0 ) p 0 , (T p 1 ,q s 1 )

p 1 ) θp θ /p 1 ,1 f | I p θ ,q s 0 ,s 1 ,θ p θ for all f ∈ L q c (X + ). We write f | ((T p 0 ,q s 0 ) p 0 , (T p 1 ,q s 1 ) p 1 ) θp θ /p 1 ,1 = k∈Z 2 -kθp θ /p 1 K 2 k , f ; (T p 0 ,q s 0 ) p 0 , (T p 1 ,q s 1 ) p 1 = k∈Z 2 -kθp θ /p 1 inf f =ϕ 0 +ϕ 1 ||ϕ 0 || p 0 T p 0 ,q s 0 + 2 k ||ϕ 1 || p 1 T p 1 ,q s 1 = k∈Z 2 -kθp θ /p 1 inf f =ϕ 0 +ϕ 1 ˆX A q (V -s 0 ϕ 0 )(x) p 0 + 2 k A q (V -s 1 ϕ 1 )(x) p 1 dµ(x) ≥ k∈Z 2 -kθp θ /p 1 ˆX inf f =ϕ 0 +ϕ 1 A q (V -s 0 ϕ 0 )(x) p 0 + 2 k A q (V -s 1 ϕ 1 )(x) p 1 dµ(x) = k∈Z 2 -kθp θ /p 1 ˆX K 2 k , 1 Γ(x) f (x); L q s 0 (X + ) p 0 , L q s 1 (X + ) p 1 dµ(x) (30) = ˆX 1 Γ(x) f | (L q s 0 (X + ) p 0 , L q s 1 (X + ) p 1 ) θp θ /p 1 ,1 dµ(x) ≃ ˆX 1 Γ(x) f | (L q s 0 (X + ), L q s 1 (X + )) θ,p θ p θ dµ(x) (31) = f | I p θ ,q s 0 ,s 1 ,θ p θ
where again the infima are taken over cylindrically supported ϕ 0 and ϕ 1 . The equality (30) is due to Lemma 2.11. The equivalence (31) follows from the power theorem. This completes the proof of Proposition 2.13.

As a corollary, we obtain half of the desired interpolation result.

Corollary 2.14. Let all numerical parameters be as in the statement of Theorem 2.9, and suppose that X is AD-regular of dimension n and unbounded.

(32) (T p 0 ,q s 0 , T p 1 ,q s 1 ) θ,p θ ֒→ Z p θ ,q s θ (X; c 0 , c 1 ). Proof. This follows from Theorem 2.7, Remark 2.8, and the density of L q c (X + ) in (T p 0 ,q s 0 , T p 1 ,q s 1 ) θ,p θ (which follows from the fact that L q c (X + ) is dense in both T p 0 ,q s 0 and T p 1 ,q s 1 , which is due to Lemma 1.4). We now prove the reverse containment in the Euclidean case. This rests on a dyadic characterisation of the spaces

Z p,q s (R n ; c 0 , c 1 ). A standard (open) dyadic cube is a set Q ⊂ R n of the form (33) Q = n i=1 (2 k x i , 2 k (x i + 1))
for some k ∈ Z and x ∈ Z n . For Q of the form (33) we set ℓ(Q) := 2 k (the sidelength of Q), and we denote the set of all standard dyadic cubes by D. For every Q ∈ D we define the associated Whitney cube

Q := Q × (ℓ(Q), 2ℓ(Q)),
and we define G := {Q : Q ∈ D}. We write R n+1

+ := (R n ) + = R n × (0, ∞). Note that G is a partition of R n+1 + up to a set of measure zero.
The following proposition is proven by a simple covering argument.

Proposition 2.15. Let p, q ∈ (0, ∞), s ∈ R, c 0 > 0 and c 1 > 1. Then for all f ∈ L 0 (R n+1 + ), ||f || Z p,q s (R n ;c 0 ,c 1 ) ≃ c 0 ,c 1   Q∈G ℓ(Q) n(1-ps) [|f | q ] p/q Q   1/p
, where

[|f | q ] Q := ˆˆQ |f (y, t)| q dy dt.
As a consequence, we gain a convenient embedding.

Corollary 2.16. Suppose q ∈ (0, ∞), p ∈ (0, q], and s ∈ R. Then

Z p,q s (R n ) ֒→ T p,q s (R n ). Proof. We have ||f || T p,q s (R n ) ≃ ˆRn ˆˆΓ(x) |t -ns f (y, t)| q dy dt t n+1 p/q dx 1/p ≤    ˆRn   Q∈G 1 Q∩Γ(x) =∅ (Q) ˆˆQ |t -ns f (y, t)| q dy dt t n+1   p/q dx    1/p ≃    ˆRn   Q∈G 1 Q∩Γ(x) =∅ (Q)ℓ(Q) -nsq [|f | q ] Q   p/q dx    1/p ≤   ˆRn Q∈G 1 Q∩Γ(x) =∅ (Q)ℓ(Q) -nps [|f | q ] p/q Q dx   1/p (34) =   Q∈G ℓ(Q) -nps [|f | q ] p/q Q |{x ∈ R n : Γ(x) ∩ Q = ∅}|   1/p   Q∈G ℓ(Q) n(1-ps) [|f | q ] p/q Q   1/p (35)
≃ ||f || Z p,q s (X;c 0 ,c 1 ) , where (34) follows from p/q ≤ 1, (35) follows from

|{x ∈ R n : Γ(x) ∩ Q = ∅}| = |B(Q, 2ℓ(Q))| |Q| ≃ ℓ(Q) n ,
and the last line follows from Proposition 2.15. This proves the claimed embedding.

It has already been shown by Barton and Mayboroda that the Z-spaces form a real interpolation scale [START_REF] Barton | Layer potentials and boundary-value problems for second order elliptic operators with data in Besov spaces[END_REF]Theorem 4.13], in the following sense. We will stop referring to the parameters c 0 and c 1 , as Proposition 2.15 implies that the associated quasi-norms are equivalent.

Proposition 2.17. Suppose that all numerical parameters are as in the statement of Theorem 2.9. Then we have the identification

(Z p 0 ,q s 0 (R n ), Z p 1 ,q s 1 (R n )) θ,p θ = Z p θ ,q s θ (R n
). Now we know enough to complete the proof of Theorem 2.9.

Proof of Theorem 2.9. First suppose that p 0 , p 1 ∈ (0, 2]. By Corollary 2.16 we have Z p j ,q s j (R n ) ֒→ T p j ,q s j (R n ), for j = 0, 1, and so

(Z p 0 ,q s 0 (R n ), Z p 1 ,q s 1 (R n )) θ,p θ ֒→ (T p 0 ,q s 0 (R n ), T p 1 ,q s 1 (R n )) θ,p θ .
Therefore by Proposition 2.17 we have Z p θ ,q s θ (R n ) ֒→ (T p 0 ,q s 0 (R n ), T p 1 ,q s 1 (R n )) θ,p θ , and Corollary 2.14 then implies that we in fact have equality, Z p θ ,q s θ (R n ) = (T p 0 ,q s 0 (R n ), T p 1 ,q s 1 (R n )) θ,p θ . This equality also holds for p 0 , p 1 ∈ (1, ∞) by Theorem 2.4. By reiteration, this equality holds for all p 0 , p 1 ∈ (0, ∞). The proof of Theorem 2.9 is now complete.

Remark 2.18. This can be extended to general unbounded AD-regular spaces by establishing a dyadic characterisation along the lines of Proposition 2.15 (replacing Euclidean dyadic cubes with a more general system of 'dyadic cubes'), and then proving analogues of Corollary 2.16 and Proposition 2.17 using the dyadic characterisation. The Euclidean applications are enough for our planned applications, and the Euclidean argument already contains the key ideas, so we leave further details to any curious readers.

2.4.

Hardy-Littlewood-Sobolev embeddings. In this section we prove the following embedding theorem. Theorem 2.19 (Weighted tent space embeddings). Suppose X is doubling. Let 0 < p 0 < p 1 ≤ ∞, q ∈ (0, ∞] and s 0 > s 1 ∈ R. Then we have the continuous embedding T p 0 ,q s 0 ֒→ T p 1 ,q s 1 whenever s 1 -s 0 = δ p 0 ,p 1 . Furthermore, when p 0 ∈ (0, ∞], q ∈ (1, ∞), and α > 0, we have the embedding T p 0 ,q s 0 ֒→ T ∞,q s 1 ;α whenever (s 1 + α) -s 0 = δ p 0 ,∞ .

These embeddings can be thought of as being of Hardy-Littlewood-Sobolev-type, in analogy with the classical Hardy-Littlewood-Sobolev embeddings of homogeneous Triebel-Lizorkin spaces (see for example [START_REF] Jawerth | Some observations on Besov and Lizorkin-Triebel spaces[END_REF]Theorem 2.1]).

The proof of Theorem 2.19 relies on the following atomic estimate. Note that no geometric assumptions are needed here.

Lemma 2.20. Let 1 ≤ p ≤ q ≤ ∞ and s 0 > s 1 ∈ R with s 1 -s 0 = δ 1,p . Suppose that a is a T 1,q s 0 -atom. Then a is in T p,q s 1 , with ||a|| T p,q s 1 ≤ 1.

Proof. Suppose that the atom a is associated with the ball B ⊂ X. When p = ∞, using the fact that B(x, t) ⊂ B whenever (x, t) ∈ T (B) and that -δ 1,p > 0, we have

||a|| T p,q s 1 = A q (V -s 1 a) L p (B) ≤ V -δ 1,p L ∞ (T (B)) A q (V -s 0 a) L p (B) ≤ µ(B) δ p,1 µ(B) δq,p ||a|| T q,q s 0 ≤ µ(B) δ p,1 +δq,p+δ 1,q = 1,
where we used Hölder's inequality with exponent q/p ≥ 1 in the third line.

When p = q = ∞ the argument is simpler: we have

||a|| T ∞,∞ s 1 = V -s 0 -δ 1,∞ a L ∞ (T (B)) ≤ V -δ 1,∞ L ∞ (T (B)) V -s 0 a L ∞ (T (B)) ≤ µ(B) δ ∞,1 µ(B) δ 1,∞ = 1
using the same arguments as before (without needing Hölder's inequality). Now we will prove the embedding theorem. Here is a quick outline of the proof. First we establish the first statement for p 0 = 1 and 1 < p 1 ≤ q by using part (1) of Lemma 2.20. A convexity argument extends this to 0 < p 0 < p 1 ≤ q, with q > 1. Duality then gives the case 1 < q ≤ p 0 < p 1 ≤ ∞, including when p 1 = ∞ and α = 0. A composition argument completes the proof with q > 1. Finally, we use another convexity argument to allow for q ∈ (0, 1] (with p 1 < ∞). To handle the second statement, we argue by duality again.

Proof of Theorem 2.19. The proof is split into six steps, corresponding to those of the outline above.

Step 1. First suppose that f ∈ T 1,q s 0 and 1 ≤ p 1 ≤ q. By the weighted atomic decomposition theorem, we can write f = k λ k a k where each a k is a T 1,q s 0 -atom, with the sum converging in T 1,q s 0 . By Lemma 2.20 we have

||f || T p 1 ,q s 1 ≤ ||λ k || ℓ 1 (N) .
Taking the infimum over all atomic decompositions yields the continuous embedding

(36) T 1,q s 0 ֒→ T p 1 ,q s 1 (1 < p 1 ≤ q ≤ ∞, s 1 -s 0 = δ 1,p 1 ).
Step 2. Now suppose 0 < p 0 < p 1 ≤ q, s 1 -s 0 = δ p 0 ,p 1 , and f ∈ T p 0 ,q s 0 . Using (36) and noting that q/p 0 > 1 and

p 0 s 1 -p 0 s 0 = p 0 δ p 0 ,p 1 = δ 1,p 1 /p 0 , we have ||f || T p 1 ,q s 1 = f p 0 | T p 1 /p 0 ,q/p 0 p 0 s 1 1/p 0 f p 0 | T 1,q/p 0 p 0 s 0 1/p 0 = ||f || T p 0 ,q s 0
, which yields the continuous embedding (37)

T p 0 ,q s 0 ֒→ T p 1 ,q s 1 (0 < p 0 < p 1 ≤ q ≤ ∞, q > 1, s 1 -s 0 = δ p 0 ,p 1 ).

Step 3. We now use a duality argument. Suppose 1 < q ≤ p 0 < p 1 ≤ ∞. Define π 0 := p ′ 1 , π 1 := p ′ 0 , ρ := q ′ , σ 0 := -s 1 , and

σ 1 := -s 0 , with s 1 -s 0 = δ p 0 ,p 1 . Then σ 1 -σ 0 = -s 0 + s 1 = δ p 0 ,p 1 = δ π 0 ,π 1 ,
and so (37) gives the continuous embedding

T π 0 ,ρ σ ֒→ T π 1 ,ρ σ 1 . Taking duals results in the continuous embedding (38) T p 0 ,q s 0 ֒→ T p 1 ,q s 1 (1 < q ≤ p 0 < p 1 ≤ ∞, s 1 -s 0 = δ p 0 ,p 1 ).
Step 4. Now suppose that 0 < p 0 ≤ q ≤ p 1 ≤ ∞ and q > 1, again with s 1 -s 0 = δ p 0 ,p 1 . Then combining (37) and (38) gives continuous embeddings (39) T p 0 ,q s 0 ֒→ T q,q s 0 +δp 0 ,q ֒→ T p 1 ,q s 0 +δp 0 ,q +δq,p 1 = T p 1 ,q s 1 .

Step 5. Finally, suppose q ≤ 1, and choose M > 0 such that q/M > 1. Then using a similar argument to that of Step 2, with Ms 1 -Ms

0 = Mδ p 0 ,p 1 = δ p 0 /M,p 1 /M , ||f || T p 1 ,q s 1 = f M | T p 1 /M,q/M M s 1 1/M f M | T p 0 /M,q/M M s 0 1/M = ||f || T p 0 ,q s .
All possible positions of q relative to 0 < p 0 < p 1 ≤ ∞ have thus been covered, so the proof of the first statement is complete.

Step 6. For the second statement, we let (s 1 + α) -s 0 = δ p 0 ,∞ , and first we suppose that p

0 ∈ (1, ∞]. Let π 0 := (1 + α) -1 ∈ (0, 1), π 1 := p ′ 0 ∈ (1, ∞], ρ = q ′ , σ 0 = -s 1 , σ 1 = -s 0 .
Then α = δ 1,π 0 = δ p 1 ,∞ and so we have

σ 1 -σ 0 = δ p 0 ,∞ -α = δ 1,π 1 -δ 1,π 0 = δ π 0 ,π 1 , which yields T π 0 ,ρ σ 0 ֒→ T π 1 ,ρ σ 1 .

Taking duals yields

T p 0 ,q s 0 ֒→ T ∞,q s 1 ,α , which completes the proof when p 0 ∈ (1, ∞]. One last convex reduction argument, as in Step 2, completes the proof.

We remark that this technique also yields the embedding T ∞,q s 0 ,α 0 ֒→ T ∞,q s 1 ,α 1 when (s 1 + α 1 ) -(s 0 + α 0 ) = 0, s 0 > s 1 , and 0 ≤ α 0 < α 1 .

Remark 2.21. The embeddings of Theorems 2.19, at least for p, q ∈ (1, ∞), also hold with Z p,q s replacing T p,q s on either side (or both sides) of the embedding. This can be proven by writing Z p,q s as a real interpolation space between tent spaces T p,q s with p near p and s near s, applying the tent space embedding theorems, and then interpolating again. These embeddings can also be proven 'by hand', even for p, q ≤ 1. We leave the details to any curious readers.

Deferred proofs

3.1. T p,∞ -L ∞ estimates for cylindrically supported functions. The following lemma, which extends [1, Lemma 3.3] to the case q = ∞, is used in the proof that T p,∞ is complete (see Proposition 1.4). Lemma 3.1. Suppose that X is doubling and let K ⊂ X + be cylindrical. Then for all

p ∈ [1, ∞], ||1 K f || T p,∞ K ||f || L ∞ (K) K ||f || T p,∞ .
Proof. When p = ∞ this reduces to

||1 K f || L ∞ (X + ) = ||f || L ∞ (K) ≤ ||f || L ∞ (X + ) ,
which is immediate. Thus it suffices to prove the result for p = 1, for the general case will then follow by interpolating between the L 1 (K) → L 1 (X) and L ∞ (K) → L ∞ (X) boundedness of the sublinear operator A ∞ . Write K ⊂ B K × (κ 0 , κ 1 ) for some ball

B K = B(c K , r K ) ⊂ X and 0 < κ 0 < κ 1 < ∞. To prove that ||1 K f || T 1,∞ K ||f || L ∞ (K) , observe that ||1 K f || T 1,∞ ≤ ||f || L ∞ (K) µ{x ∈ X : Γ(x) ∩ K = ∅} ≤ ||f || L ∞ (K) V (c K , r K + κ 1 ) because if x / ∈ B(c K , r K + κ 1 ) then Γ(x) ∩ (B K × (κ 0 , κ 1 )) = ∅. Note also that V (c K , r K + κ 1
) is finite and depends only on K. Now we will prove that ||f || L ∞ (K) K ||f || T 1,∞ . First note that the doubling property implies that for all R > 0 and for all balls B ⊂ X,

(40) inf x∈B µ(B(x, R)) X,R,r B µ(B). Indeed, if x ∈ B and R ≤ 2r B then µ(B) ≤ µ(B(x, R(2r B R -1 ))) X (2r B R -1 ) n µ(B(x, R)).
where n ≥ 0 is the doubling dimension of X. If R > 2r(B) then since 2r B R -1 < 1, we have µ(B) ≤ µ(B(x, R)).

Let (x j ) j∈N be a countable dense subset of B K . Then we have

K = j∈N (Γ(x j ) + κ 0 ) ∩ K.
By definition the set {(y, t)

∈ K : |f (y, t)| > 2 -1 ||f || L ∞ (K) } has positive measure, so there exists j ∈ N such that |f (y, t)| > 2 -1 ||f || L ∞ (K)
for (y, t) in some subset of (Γ(x j ) + κ 0 ) ∩ K with positive measure. Since (Γ(x j ) + κ 0 ) ∩ K ⊂ Γ(x) ∩ K for all x ∈ B(x j , κ 0 ), we have that A ∞ (f )(x) ≥ 2 -1 ||f || L ∞ (K) for all x ∈ B(x j , κ 0 ). Therefore, using (40),

||A ∞ f || L 1 (X) ≥ 1 2 µ(B(x j , κ 0 )) ||f || L ∞ (K) X,K µ(B K ) ||f || L ∞ (K) ≃ K ||f || L ∞ (K) .
This completes the proof of the lemma.

3.2. T p,∞ atomic decomposition. As stated above, the atomic decomposition theorem for T p,∞ can be proven by combining the arguments of Coifman-Meyer-Stein (who prove the result in the Euclidean case) and Russ (who proves the atomic decomposition of T p,2 (X) for 0 < p ≤ 1 when X is doubling). First we recall a classical lemma (see for example [21, Lemma 2.2]), which combines a Vitali-type covering lemma with a partition of unity. This is proven by combining the Vitali-type covering of Coifmann-Weiss [10, Théorème 1.3] with the partition of unity of Macías-Segovia [START_REF] Macías | Decomposition into atoms of distributions on spaces of homogeneous type[END_REF]Lemma 2.16]. Lemma 3.2. Suppose that X is doubling, and let O be a proper subset of X of finite measure. For all x ∈ X write r(x) := dist(x, O c )/10. Then there exists M > 0, a countable indexing set I, and a collection of points {x i } i∈I such that

• O = ∪ i∈I B(x i , r(x i )),

• if i, j ∈ I are not equal, then B(x i , r(x i )/4) and B(x j , r(x j )/4) are disjoint, and • for all i ∈ I, there exist at most M indices j ∈ I such that B(x j , 5r(x j )) meets B(x i , 5r(x i )). Moreover, there exist a collection of measurable functions {ϕ i : X → [0, 1]} i∈I such that

• supp ϕ i ⊂ B(x i , 2r(x i )),

• i ϕ i = 1 O (for each x ∈ X the sum i ϕ i (x) is finite due to the third condition above).

p k∈Z ˆ2k 2 k-1 t p-1 µ({x ∈ X : A ∞ f (x) > t}) dt = ||A ∞ f || p L p (X)
= ||f || T p,∞ , using doubling in (43) and pairwise disjointness of the balls B(x k i , r(x k i )/4) in (44). This completes the proof in the case that µ(X) = ∞.

Case 2: µ(X) < ∞. In this case we may have O k = X for some k ∈ Z, so we cannot apply Lemma 3.2 as before. One can follow the argument of Russ [21, page 131], which shows that the partition of unity is not required for such k. With this modification, the argument of the previous case still works. We omit the details.

k∈Z 1 T

 1 Now we can follow a simplified version of the argument of Russ, which is essentially the argument of Coifman-Meyer-Stein with the partition of unity of Lemma 3.2 replacing the use of the Whitney decomposition.Proof of Theorem 1.6, with q = ∞. Suppose f ∈ T p,∞ , and for each k ∈ Z define the setO k := {x ∈ X : A ∞ f (x) > 2 k }.The sets O k are open by lower semicontinuity of A ∞ f (Lemma 1.1), and the function f is essentially supported in ∪ k∈Z T (O k ) \ T (O k+1 ). Thus we can write (41) f = (O k )\T (O k+1 ) f. Case 1: µ(X) = ∞. In this case we must have µ(O k ) < ∞ for each k ∈ Z, for otherwise we would have ||A ∞ f || L p (X) = ∞ and thus f / ∈ T

  p,∞ . Hence for each k ∈ Z there exist countable collections of points {x k i } i∈I k ⊂ O k and measurable functions {ϕ k i } i∈I k as in Lemma 3.2. Combining (41) with i∈I k ϕk i = 1 O k and T (O k ) ⊂ O k × R + , we can write f (y, t) = T (O k )\T (O k+1 ) (y, t)f (y, t) |f (y, t)| ≤ 2 k+1 , the second inequality following from T (O k+1 ) = X + \ (∪ x / ∈O k+1 Γ(x)) and the fact that |f (y, t)| ≤ A ∞ f (x) ≤ 2 k+1 for all x / ∈ O k+1 and (y, t) ∈ Γ(x). Define a k i := 2 -(k+1) µ(B k i ) -1/p ãk i , where B k i := B(x k i , 14r(x k i )). We claim that a k i is a T p,∞ -atom associated with the ball B k i . The estimate (42) immediately implies the size conditiona k i T ∞,∞ ≤ µ(B k i ) δp,∞ , so we need only show that a k i is essentially supported in T (B k i ). To show this, it is sufficient to show that if y ∈ B(x k i , 2r(x kwhere the last line follows from u / ∈ O k and d(y, (O k ) c ) ≥ t. Since z / ∈ B k i and ε > 0 were arbitrary, this shows that d(y, (B k i ) c ) ≥ t as required, which proves that a k i is a T p,∞ -atom associated with B k i .

	We estimate
		|λ k i | p =	2 (k+1)p	µ(B k i )
	(43) (44)	k∈Z i∈I k k∈Z ϕ k X i (y)1 = k∈Z i∈I k 2 (k+1)p i∈I k k∈Z i∈I k ãk k∈Z i∈I k i (y, t). ≤ 2 (k+1)p µ(O k ) µ(B(x k i , r(x k i )/4))
	Note that	k∈Z
	(42)	ãk i L ∞ (X + ) ≤ ess sup (y,t) / ∈T (O k+1 )
				∈ O k
	such that
		d(x k i , u) < d(x k i , (O k ) c ) + ε = 10r(x k i ) + ε.
	Then we have
		d(y, z) + ε ≥ d(z, x k i ) -d(x k i , y) + ε
		≥ 12r(x k i ) + ε
		= 2r(x k i ) + 10r(x k i ) + ε
		> d(y, x k i ) + d(x k i , u)
		≥ d(y, u)
		≥ t,
	Thus we have
		f (y, t) =	λ k i a k i ,
			k∈Z i∈I k
	where
		λ k i = 2 k+1 µ(B k i ) 1/p .
	It only remains to show that
			|λ k i | p ||f || p T p,∞ .
		k∈Z i∈I k

i )) and d(y, (O k ) c ) ≥ t, then d(y, (B k i ) c ) ≥ t. Suppose z / ∈ B k i (such a point exists because µ(B k i ) < µ(X) = ∞), ε > 0 and u /

The cases where q = ∞ are not covered there. The same proof works-the only missing ingredient is Lemma 3.1, which we defer to the end of the article.

Harboure, Torrea, and Viviani [12] avoid this problem by embedding T 1 into a vector-valued Hardy space H 1 . If we were to extend this argument we would need identifications of quasi-Banach real interpolants of certain vector-valued Hardy spaces H p for p ≤ 1, which is very uncertain terrain (seeBlasco and Xu [7]).


Florent Malrieu

Tran Hoa Phu

Lotka-Volterra with randomly fluctuating environments: a full description

In this note, we study the long time behavior of Lotka-Volterra systems whose coefficients vary randomly. Benaïm and Lobry established that randomly switching between two environments that are both favorable to the same species may lead to four different regimes: almost sure extinction of one of the two species, random extinction of one species or the other and persistence of both species. Our purpose here is to provide a complete description of the model. In particular, we show that any couple of environments may lead to the four different behaviours of the stochastic process depending on the jump rates.

Introduction

For a given set of positive parameters ε = (a, b, c, d, α, β), consider the Lotka-Volterra differential system in R 2 + , is given by    x = αx(1 -ax -by) y = βy(1 -cx -dy) (x 0 , y 0 ) ∈ R 2

+

We denote by F ε the associated vector field: (x , y ) = F ε (x, y). Let us note already that when a < c and b < d, the point (1/a, 0) attracts any path starting in (0, +∞) 2 . We say that the environment is favorable to species x. Similarly, when a > c and b > d, the point (0, 1/d) attracts any path starting in (0, +∞) 2 . We say that the environment is favorable to species y. See [?] for a detailed presentation of the four generic configurations. The environment is said to be of Consider two such systems ε 0 = (a 0 , b 0 , c 0 , d 0 , α 0 , β 0 ) and ε 1 = (a 1 , b 1 , c 1 , d 1 , α 1 , β 1 ) and introduce the random process {(X t , Y t , I t )} on R × R × {0, 1} obtained by switching between these two deterministic dynamics, at rates λ 0 , λ 1 . More precisely, we consider the Markov process driven by the following generator

•
Lf (z, i) = F i (z) • ∇ z f (z, i) + λ i (f (z, 1 -i) -f (z, i)), (z, i) ∈ R 2 × {0, 1}.
Equivalently, (I t ) t≥0 is a Markov process on {0, 1} with jump rate λ 0 and λ 1 , that is

P(I t+s = 1 -i|I t = i, F t ) = λ i s + o(s),
where F t is the sigma field generated by {I u , u ≤ t}. Finally, (X t , Y t ) is solution of

(X t , Y t ) = F ε I t (X t , Y t ).
This process on R 2 × {0, 1} has already been studied in [?, ?]. It belongs to the class of the piecewise deterministic Markov processes introduced by Davis [?]. See also [?] for a recent review of the application areas of such processes. Let us introduce the invasion rates of species x and y defined in [?] as

Λ y = β 0 (1 -c 0 x)µ(dx, 0) + β 1 (1 -c 1 x)µ(dx, 1), Λ x = α 0 (1 -b 0 y) µ(dy, 0) + α 1 (1 -b 1 x) µ(dy, 1),
where µ is the invariant probability measure of (X t , I t ) associated to equation:

X t = α It X t (1 -a It X t ),
and µ is the invariant probability measure of (Y t , I t ) associated to equation:

Y t = β It Y t (1 -d It Y t ).
The meaning of Λ y is the following: when species y is close to extinction, species x behaves approximately as (X t , 0) = F ε I t (X t , 0) and Λ y is the growth rate of species y with respect to invariant measure µ of (X, I). Note that the invasion rates depend on the jump rates (λ 0 , λ 1 ) ∈ (0, +∞) 2 . For every (λ 0 , λ 1 ) ∈ (0, +∞) 2 , we have two parametrizations of these jump rates:

(s, t) ∈ [0, 1] × (0, +∞) : st = λ 0 , (1 -s)t = λ 1 . (u, v) ∈ [0, 1] × (0, +∞) : uv = λ 0 /α 0 , (1 -u)v = λ 1 /α 1 .
The change of parameters (u, v) = ξ(s, t) is triangular in the sense that u only depends on s

(u, v) = ξ(s, t) = sα 1 (1 -s)α 0 + sα 1 , t α 0 α 1 ((1 -s)α 0 + sα 1 ) .
Let us denote the invasion rates in the (u, v) coordinates by

Λx (u, v) = Λ x (ξ -1 (u, v)) and Λy (u, v) = Λ y (ξ -1 (u, v)).
It is established in [?] that signs of Λx and Λy determine the long time behavior of (X t , Y t ).

Λy > 0 Λy < 0 Λx > 0 persistence of the two species extinction of species y Λx < 0 extinction of species x extinction of species x or y Moreover, in [?] it is shown that two environments of Type 1 may lead to four regimes for the stochastic process. This surprising result is reminiscent of switched stable linear ODE studied in [?, ?].

A fundamental property of the model is that, for all 0 ≤ s ≤ 1, the vector field (1-s)F ε 0 +sF ε 1 is the Lotka-Volterra system associated to the environment ε s = (a s , b s , c s , d s , α s , β s ) with

α s = sα 1 + (1 -s)α 0 , a s = sα 1 a 1 + (1 -s)α 0 a 0 α s , b s = sα 1 b 1 + (1 -s)α 0 b 0 α s , (1.1)
β s = sβ 1 + (1 -s)β 0 , c s = sβ 1 c 1 + (1 -s)β 0 c 0 β s , d s = sβ 1 d 1 + (1 -s)β 0 d 0 β s . (1.2) Set I = {0 ≤ s ≤ 1 : a s > c s } and J = {0 ≤ s ≤ 1 : b s > d s }.
We denote by Ĩ the image of I for the other parametrization.

Remark 1.1. As noticed in [?], if ε 0 and ε 1 are of Type 1 then I or J may generically be empty or an open interval which closure is contained in (0, 1).

Let us recall below the key result in [?] about the expression of the invasion rates.

Lemma 1.2. [?, Lemma 1.2] Assume that ε 0 and ε 1 are of Type 1 and, w.l.g., a 0 < a 1 . The quantity Λy can be rewritten as:

Λy (u, v) = 1 (a 1 -a 0 )( 1 α 0 (1 -u) + 1 α 1 u) E[φ(U u,v )]
where φ : [0, 1] → R is defined by

φ(y) = (a 0 + (a 1 -a 0 )y)P ( 1 a 0 + (a 1 -a 0 )y ),
where

P (x) = β 1 α 1 (1 -c 1 x)(1 -a 0 x) - β 0 α 0 (1 -c 0 x)(1 -a 1 x) a 1 -a 0 |a 1 -a 0 | , (1.3)
and U u,v is a Beta distributed Beta(uv, (1 -u)v) random variable. Moreover, φ has the following properties:

• If I is empty then φ is nonpositive.

• If I is nonempty (I = (u 1 , u 2 )) then φ is concave, negative on (0, u 1 ) ∪ (u 2 , 1) and positive on Ĩ = (u 1 , u 2 ).
Our first result is the precise study of the properties of Λx and Λy with two environments ε 0 , ε 1 that are respectively of Type 1 and Type 2. In particular, we describe the regions where Λx and Λy are positive.

Theorem 1.3. (Shape of the regions). Assume that ε 0 and ε 1 are respectively of Type 1 and Type 2. Then, there exists a function u → v y (u) from (0, 1) → [0, ∞], such that Λy (u, v) < 0 when v < v y (u) and Λy (u, v) > 0 when v > v y (u). Let a be the coefficient of second degree of polynomial P given by (??).

If a < 0, there exists 0 < α < α < 1 such that v y is infinite on [0, α], is decreasing and continuous on (α, α), tends to +∞ at α, tends to 0 at α and is equal to 0 on [α, 1].

If a > 0, there exists 0 < α < α < 1 such that v y is equal to 0 on [0, α], is increasing and continuous on (α, α), tends to 0 at α, tends to +∞ at α, and is infinite on [α, 1].

Moreover, α and α are explicit.

The second result is the following theorem.

Theorem 1.4. For any (i, j) in {1, 2, 3, 4} 2 , there exist two environments ε 0 of Type i and ε 1 of Type j such that the associated stochastic process has four possible regimes depending on the jump rates.

The paper is organized as follows. In Section 2 we prove the properties of Λx and Λy . In Section 3 we prove Theorem ??. In Section 4 we present illustrations obtained by numerical simulation. In Section 5 we study the case when the two environments are of Type 3. Finally, in Section 6, we prove Theorem ?? providing, in each case, a good couple of environments.

Expression of invasion rates

Lemma 2.1. If ε 0 and ε 1 are respectively of Type 1 and Type 2, then Ĩ is always nonempty and there exists 0 < α < 1 (depends on α i , β i , a i , c i ) such that Ĩ = (α, 1].

Proof. Set R = β 0 α 1 α 0 β 1 , u = sα 1 α s , A = (a 1 -a 0 )(R-1), B = (2a 0 -c 0 -a 1 )R+(c 1 -a 0 ), C = (c 0 -a 0 )R.
For any s ∈ (0, 1), we get that

c s -a s = Au 2 + Bu + C R(1 -u) + u
where a s and c s are given by (??) and (??). Set

T (u) = Au 2 + Bu + C ∀u ∈ [0, 1].
We easily get

T (0) = C = (c 0 -a 0 )R > 0, T (1) = A + B + C = c 1 -a 1 < 0.
Because T is a second degree polynomial with T (0) > 0 and T (1) < 0, we conclude that

T (u) < 0 ⇔ u > α = -B - √ B 2 -4AC 2A . Therefore u ∈ Ĩ ⇔ T (u) < 0 ⇔ u > α ⇔ u ∈ (α, 1]. As a consequence, Ĩ = (α, 1].
Proposition 2.2. The map Λy (u, v) satisfies the following properties:

For all u ∈ [0, 1] lim v→∞ Λy (u, v) = β u (1 - c u a u )      > 0 if u ∈ Ĩ = (α, 1], = 0 if u ∈ ∂ Ĩ = {α}, < 0 if u ∈ (0, 1) \ Ĩ = [0, α),
and

lim v→0 Λy (u, v) = 1 1 α 0 (1 -u) + 1 α 1 u β 1 α 1 (1 - c 1 a 1 ) - β 0 α 0 (1 - c 0 a 0 ) u + β 0 α 0 (1 - c 0 a 0 ) . (2.1)
Proof. The proposition is obtained by changing variables (s, t) ←→ (u, v) from [?, Prop. 2.3].

Proposition 2.3. There exists 0 < α < 1 such that lim v→0 Λy (u, v) > 0 if u > α and lim v→0 Λy (u, v) < 0 if u < α.

Proof. The limit in (??) has the same sign than

g(u) = β 1 α 1 (1 - c 1 a 1 ) - β 0 α 0 (1 - c 0 a 0 ) u + β 0 α 0 (1 - c 0 a 0 ), ∀u ∈ [0, 1].
We get

g(0) = β 0 α 0 (1 - c 0 a 0 ) < 0 and g(1) = β 1 α 1 (1 - c 1 a 1 ) > 0.
Since g is a linear function, α is the unique zero of g and the result is clear.

Proposition 2.4. Let a be the coefficient of degree 2 of polynomial P given by (??)

a = β 1 α 1 c 1 a 0 - β 0 α 0 c 0 a 1 a 1 -a 0 |a 1 -a 0 | .
If a < 0 (resp. a > 0 or a = 0) then α < α (resp. α > α or α = α).

Proof. By symmetry we only consider the case a < 0. Without loss of generality, we assume that a 1 > a 0 and a becomes:

a = β 1 α 1 c 1 a 0 - β 0 α 0 c 0 a 1 .
To prove that α < α, it is sufficient to prove Aα 2 + Bα + C < 0. Since, by definition of α,

β 1 α 1 1 - c 1 a 1 - β 0 α 0 1 - c 0 a 0 α + β 0 α 0 1 - c 0 a 0 = 0,
we get, multiplying by a 0 a 1 α 1 /β 1 , that

(a 0 a 1 -c 1 a 0 -Ra 1 a 0 + Ra 1 c 0 )α + Ra 1 (a 0 -c 0 ) = 0. (2.2)
Replacing α by its expression in (??), we get:

Aα 2 + Bα + C = R(a 1 -c 1 )(a 0 -a 1 )(a 0 -c 0 )(a 0 c 1 -Ra 1 c 0 ) (a 0 a 1 -a 0 c 1 -Ra 0 a 1 + Ra 1 c 0 ) 2 . Since c 0 > a 0 , a 1 > c 1 , a 1 > a 0 and a 0 c 1 -Ra 1 c 0 = α 1 β 1 a < 0, we conclude Aα 2 + Bα + C < 0.
3 Shape of the positivity region

Recall a = β 1 α 1 a 0 c 1 - β 0 α 0 a 1 c 0 a 1 -a 0 |a 1 -a 0 |
is the coefficient of degree 2 of polynomial P given by (??).

Lemma 3.1. [?, Lemma 4.1] Assume ε 0 and ε 1 are of Type 1. If Ĩ is nonempty, then the map

(u, v) → E[φ(U u,v )
] is increasing in v and concave in u.

Remark 3.2. In Benaïm and Lobry's case, if I is nonempty, φ is concave and the parameter a is always negative. In the present case, a may be negative, positive or zero. Therefore, we have the following lemma.

Lemma 3.3. Assume ε 0 and ε 1 are respectively of Type 1 and Type 2, then the shape of φ depends on the sign of a:

• If a is negative, then φ is strongly concave and (u, v) → E[φ(U u,v )] is increasing in v and concave in u.

• If a is positive, then φ is strongly convex and

(u, v) → E[φ(U u,v )] is decreasing in v and convex in u.
• If a is zero, then φ is linear and

(u, v) → E[φ(U u,v )] is constant in v and linear in u.
Proof. This is a straightforward adaptation of [?, Lem 4.1].

Let us conclude this section with the proof of Theorem ??.

Proof of Theorem ??. We consider only the case a < 0. Set K = (α, α). We know clearly that v → Λy (u, v) admits:

• negative limits at 0 and ∞ if u ∈ [0, α),

• positive limits at 0 and ∞ if u ∈ (α, 1],

• a negative limit at 0 and a positive limit at ∞ if u ∈ (α, α).

The fact that v → Λy (u, v) is increasing justifies the existence of v y , and we have

Λy (u, v) = 0 ⇔ u ∈ K, v = v y (u).
Let us prove that v y is decreasing in K. Let δ 1 < δ 2 be two points in K. Choose any δ 3 ∈ (α, 1), we get Λy (δ 1 , v y (δ 1 )) = 0 and Λy (δ 3 , v y (δ 1 )) > 0. Since Λx (•, v y (δ 1 )) is concave and δ 1 < δ 2 < δ 3 we get Λy (δ 2 , v y (δ 1 )) > 0. Since Λy (δ 2 , •) is increasing, we obtain v y (δ 2 ) < v y (δ 1 ).

The continuity of v y on K is a straightforward consequence of the continuity of the function Λy , which is obvious from the expression (??).

Let us show v y tends to ∞ on α.

Let {u n } ⊂ K : u n ↓ α. Since v y is decreasing in K, we get v y (u n ) ↑ v ∈ [0, ∞]. If v is finite, since the zero set of Λy is closed, by continuity, α ∈ K (impossible). So v y (u n ) ↑ ∞.
Let us prove v y tends to 0 on α.

Let {u n } ⊂ K : u n ↑ α. Since v y is decreasing in K, we get v y (u n ) ↓ ∈ [0, ∞). If > 0, since u n < α, we obtain Λy (u n , /2) < 0 ∀n. Therefore 0 < Λy (α, /2) = lim
n→∞ Λy (u n , /2) ≤ 0 (impossible). As a consequence, = 0 and v y (u n ) ↓ 0.

Numerical illustrations

Recall that for all u ∈ [0, 1], v y (u) and v x (u) are the unique respective solutions of Λy (u, v) = 0 and Λx (u, v) = 0.

We now consider, for a varying parameter ρ, the environments Thanks to [?], these plots give us information about how many regimes we can observe when the jump rates are modified. For example, the plot for ρ = 10 has three regimes: extinction of x (on the right of the green curve), persistence (between the green and blue curves) and extinction of y (on the left of the blue curve). For ρ = 9, there is an additional zone (above the red curve and below the blue curve) that corresponds to jump rates leading to random extinction of a species.

ε 0 = (1,

Switching between two persistent Lotka-Volterra systems

Let us assume that ε 0 and ε 1 are of Type 3. In this case, one can easily get that extinction of species y is not possible if u is to close of 0 or 1; in other words, [0

, 1] \ Ĩ is either empty or is an open interval which closure is contained in [0, 1]. Recall R = β 0 α 1 α 0 β 1 , A = (a 1 -a 0 )(R -1), B = (2a 0 -c 0 -a 1 )R + (c 1 -a 0 ), C = (c 0 -a 0 )R.
Then, we get that Figure ?? provides the shape of v x and v y for the environments ε 0 = (6, 1, 4, 2, 1, 5) and ε 1 = (3, 3, 2, 5.5, 5, 1). Once again, the switched process has four regimes depending on the jump rates.

[0, 1] \ Ĩ = ∅ ⇔          A < 0 ∆ = B 2 -4AC > 0 0 < -B - √ ∆ 2A < 1. Moreover, if [0, 1] \ Ĩ is nonempty, then the map (u, v) → E[φ(U u,v )] is (
Remark 5.1. We see a surprising result : although both vector fields are persistent, the stochastic process may lead to the extinction of one of the two species. 6 General case: proof of Theorem ??

The following array presents, for any couple of types, an example of two environments that are associated to a stochastic process with four regimes depending on the jump rates. Acknowledgements. This work has been written during the stay of Tran Hoa Phu in Tours for his intership of the French-Vietnam Master in Mathematics. We acknowledge financial support from the French ANR project ANR-12-JS01-0006-PIECE.

Type 1 :

 1 if a < c, b < d (favorable to species x) • Type 2: if a > c, b > d (favorable to species y) • Type 3: if a > c, b < d (persistence) • Type 4: if a < c, b > d (extinction of species x or y depending on the starting point)

Figure 1 :

 1 Figure1: The blue curve is the graph of v y (it does not depend on ρ); the green and red curves are v x for the environments given in (??) with ρ = 10 and ρ = 9 respectively.

Figure ? ?

 ? Figure ?? represents the "critical" functions v y and v x for different choices of the environments. Thanks to [?], these plots give us information about how many regimes we can observe when the jump rates are modified. For example, the plot for ρ = 10 has three regimes: extinction of x (on the right of the green curve), persistence (between the green and blue curves) and extinction of y (on the left of the blue curve). For ρ = 9, there is an additional zone (above the red curve and below the blue curve) that corresponds to jump rates leading to random extinction of a species.

  strictly) decreasing in v and convex in u. This is a straightforward adaptation of Lemma 4.1 in [?].

Figure 2 :

 2 Figure 2: Graph of v y (blue curve) and v x (red curve) for the environments ε 0 = (6, 1, 4, 2, 1, 5) and ε 1 = (3, 3, 2, 5.5, 5, 1).

1

 1 The first line has been obtained in[?]. The second line is studied in Section 2. The fifth line is studied in Section 5. The reader can easily check that the other cases correspond to Figure??.(F 0 , F 1 ) a 0 b 0 c 0 d 0 α 0 β 0 a 1 b 1 c 1 d 1 α 1 β Type

4 Figure 3 :

 43 Figure 3: Graph of v y (blue curve) and v x (red curve) for the four last cases.