Pascal Auscher

Simon Bortz
email: bortz010@umn.edu

AND Moritz Egert
email: moritz.egert@math.u-psud.fr

Olli Saari
email: saari@math.uni-bonn.de

NON-LOCAL GEHRING LEMMAS IN SPACES OF HOMOGENEOUS TYPE AND APPLICATIONS

Keywords: 2010 Mathematics Subject Classification. Primary: 30L99; Secondary: 34A08, 42B25 Gehring's lemma, (non-local) Reverse Hölder inequalities, spaces of homogeneous type, (very weak) A ∞ weights, C p weights, fractional elliptic equations, selfimprovement properties

published or not. The documents may come

Introduction

Gehring's lemma [START_REF] Gehring | The L p -integrability of the partial derivatives of a quasiconformal mapping[END_REF] establishes the open-ended property of reverse Hölder classes. If

(1.1) 1 |B| B u q dx 1/q 1 |B| B u dx
with q > 1 and all Euclidean balls B ⊂ R n , then

1 |B| B u q+ǫ dx 1/(q+ǫ) q 1 |B| B u dx
for a certain ǫ > 0 and all Euclidean balls. This self-improving property has proved to be an important tool when studying elliptic [START_REF] Elcrat | Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions[END_REF]10] and parabolic [11] partial differential equations as well as quasiconformal mappings [16]. In this case, one has to enlarge the ball in the right hand side. We come back to this. In this work, we are concerned with reverse Hölder inequalities when the righthand side is non-local. Understanding an analogue of Gehring's lemma in this generality turned out to be crucial in [START_REF] Auscher | On regularity of weak solutions to linear parabolic systems with measurable coefficients[END_REF], where we prove Hölder continuity in time for solutions of parabolic systems. The non-local nature arises from the use of halforder time derivatives. The ambient space being quasi-metric instead of Euclidean is also an assumption natural from the point of view of parabolic partial differential equations. Hence, we shall explore these non-local Gehring lemmas in spaces of homogeneous type.

It is well known that Gehring's lemma holds for the so called weak reverse Hölder inequality where the right-hand side of (1.1) is an average over a dilated ball 2B. We replace the single dilate by a significantly weaker non-local tail such as

∞ k=0 2 -k 1 |2 k B| 2 k B u dx
and certain averages over additional functions f and h that have a special meaning in applications. The main result of this paper is Theorem 3.2 asserting that a variant of Gehring's lemma, and in particular the local higher integrability of u still holds in this setting. We present a core version of the theorem already in the next section. It comes with the introduction of some necessary notation but we tried to keep things simple to give the reader a first flavor of our results. Once the strategy is in place, we discuss various consequences (Section 5), ways to generalize it (Sections 4 and 7) as well as self-improving properties for the right-hand side of the reverse Hölder inequality with tail (Section 6). We aim at covering all the aspects that usually arise from applications. Finally, we illustrate our main result by an application to regularity of solutions to the fractional divergence form equation introduced in [START_REF] Shieh | On a new class of fractional partial differential equations[END_REF] and investigated further for example in [START_REF] Schikorra | L p theory for fractional gradient PDE with V MO coefficients[END_REF]. The context of our work is the following. Gehring's lemma in a metric space endowed with a doubling measure was proved in [START_REF] Zatorska-Goldstein | Very weak solutions of nonlinear subelliptic equations[END_REF]. See also the book [START_REF] Björn | Nonlinear potential theory on metric spaces[END_REF]. By [START_REF] Macías | Lipschitz functions on spaces of homogeneous type[END_REF], every quasi-metric space carries a compatible metric structure so that Gehring's lemma also holds in that setting. However, in the case of homogeneous reverse Hölder inequalities, a very clean argument using self-improving properties of A ∞ weights was used in [START_REF] Anderson | Weak A ∞ weights and weak reverse Hölder property in a space of homogeneous type[END_REF] to give an intrinsically quasi-metric proof (see also the very closely related work [START_REF] Hytönen | Sharp reverse Hölder property for A ∞ weights on spaces of homogeneous type[END_REF]). We do not attempt to review the literature in the Euclidean n-space, but we refer to the excellent survey in [START_REF] Iwaniec | The Gehring lemma, Quasiconformal mappings and analysis[END_REF] instead. In addition, we want to point out the recent paper on Gehring's lemma for fractional Sobolev spaces [START_REF] Kuusi | Nonlocal self-improving properties[END_REF]. That paper studies fractional equations, whose solutions are self-improving in terms of both integrability and differentiability. Such phenomena are different from what we encounter here, but we found the technical part of [START_REF] Kuusi | Nonlocal self-improving properties[END_REF] very inspiring.

Among generalizations, we mention that the tails may be replaced by some supremum of averages taken over balls larger than the original ball on the lefthand side and/or that one may work on open subsets. In this way, our methods can also be applied to obtain a generalization of A ∞ weights: In [START_REF] Anderson | Weak A ∞ weights and weak reverse Hölder property in a space of homogeneous type[END_REF], a larger class of weak A ∞ weights, generalizing the one considered in [8,28] was defined and their higher (than one) integrability was proved (in spaces of homogeneous type). This class of weights, larger than the usual A ∞ Muckenhoupt class, is defined by allowing a uniform dilation of the ball in the right-hand side compared to the one on the left-hand side. Here, we show that, in fact, the dilation may be arbitrary (depending on the ball) provided it is finite. Another family of weights covered by our methods is the C p class studied in [START_REF] Muckenhoupt | Norm inequalities relating the Hilbert transform to the Hardy-Littlewood maximal function[END_REF][START_REF] Sawyer | Norm inequalities relating singular integrals and the maximal function[END_REF]. Precise definitions are given in Section 8.

Metric spaces

A space of homogeneous type (X, d, µ) is a triple consisting of a set X, a function d : X × X → [0, ∞) satisfying the quasi-distance axioms (i) d(x, y) = 0 if and only if x = y, (ii) d(x, y) = d(y, x) for all x, y ∈ X, and (iii) d(x, z) ≤ K(d(x, y) + d(y, z)) for a certain K ≥ 1 and all x, y, z ∈ X; and a Borel measure µ that is doubling in the sense that 0 < µ(B(x, 2r)) ≤ C d µ(B(x, r)) < ∞ holds for a certain C d and all radii r > 0 and centers x ∈ X. If the constant K appearing in the triangle inequality (iii) equals 1, we call (X, d, µ) a metric space with doubling measure. The topology is understood to be the one generated by the quasi-metric balls. For simplicity, we impose the additional assumption that all quasi-metric balls are Borel measurable. In general, they can even fail to be open.

The doubling condition implies there is C > 0 so that for some D > 0,

(2.1) µ(B(x, R)) µ(B(x, r)) ≤ C R r D for all x ∈ X and R ≥ r > 0. We can always take D = log 2 C d . In the following we call this number the homogeneous dimension (although there might be smaller positive numbers D than log 2 C d for which this inequality holds: our proofs work with any such D). For all these basic facts on analysis in metric spaces, we refer to the book [START_REF] Björn | Nonlinear potential theory on metric spaces[END_REF].

The following theorem is concerned with the special case of metric spaces, but it has an analogue in the general case of quasi-metric spaces, see Theorem 3.2 below.

Theorem 2.2. Let (X, d, µ) be a metric space with doubling measure. Let s, β > 0 and q > 1 be such that s < q and β ≥ D(1/s -1/q) where D is any number satisfying (2.1). Let N > 1 and let (α k ) k≥0 be a non-increasing sequence of positive numbers with α := k α k < ∞, and define

(2.3) a u (B) := ∞ k=0 α k N k B u dµ
for u ≥ 0 locally integrable and B a metric ball. Suppose that u, f, h ≥ 0 with u q , f q , h s ∈ L 1 loc (X, dµ) and A ≥ 0 is a constant such that for every ball B = B(x, R),

(2.4) B u q dµ 1/q ≤ Aa u (B) + (a f q (B)) 1/q + R β (a h s (B)) 1/s .
Then there exists p > q depending on α 0 , α, A, q, s, N and C d such that for all balls B, B u p dµ

1/p a u (NB) + (a f q (NB)) 1/q + R β (a h s (NB)) 1/s + NB f p dµ 1/p + R β NB h ps/q dµ q/sp , (2.5)
with implicit constant depending on α 0 , α, A, q, s, β, N and C d . Remark 2.6. If one assumes the sequence (α k ) k≥0 is finite, the functional is comparable to one single average on N k 0 B for some k 0 . This gives a proof of the classical Gehring lemma with dilated balls. Note the shift from N k 0 B to N k 0 +1 B in the conclusion. But well-known additional covering arguments show that the dilation factor N k 0 +1 can be changed to any number larger than 1. If one assumes

(2.7) ∃C < ∞ : ∀k ≥ 0 α k ≤ Cα k+1 ,
then it follows that a u (NB) ≤ Ca u (B) for all u ≥ 0 and all balls B. In that case, one can replace NB by B in the right-hand side of (2.5). Geometric sequences, which are typical in application, do satisfy this condition but this rules out finite sequences. Finally, note that the higher integrability of u on B depends only on the higher integrability of f and h on the first dilated ball NB.

Proof. We prove (2.5) for B = B(x 0 , R) with x 0 ∈ X and R > 0. Throughout, we reserve the symbol C for a constant that depends at most on α 0 , α, A, q, s, β, N and C d but that may vary from line to line.

Step 1. Preparation. Having fixed B, we set g q := A q R h s ½ NB with A R a constant so that for any ball B r with radius r contained in NB, we have

(2.8) r β B r h s dµ 1/s ≤ B r g q dµ 1/q and
(2.9)

NB g q dµ 1/q ≤ C 1 (NR) β NB h s dµ 1/s
for some C 1 depending only on the doubling condition, s and q. Indeed, write B r = B(x, r). As x ∈ NB, we have

NB = B(x 0 , NR) ⊂ B(x, 2NR), hence µ(NB) µ(B r ) ≤ µ(B(x, 2NR)) µ(B(x, r)) ≤ C 0 2NR r D ≤ C 0 2 D NR r β(1/s-1/q) -1
where C 0 depends only on the doubling condition. Unraveling this inequality and setting

C 1 = (C 0 2 D ) 1/q-1/s yield r β µ(B r ) 1/q-1/s ≤ C 1 (NR) β µ(NB) 1/q-1/s .
Hence, as q > s,

r β µ(B r ) 1/q-1/s B r h s dµ 1/s-1/q ≤ C 1 (NR) β µ(NB) 1/q-1/s NB h s dµ 1/s-1/q so that r β B r h s dµ 1/s = r β B r h s dµ 1/s-1/q B r h s dµ 1/q ≤ C 1 (NR) β NB h s dµ 1/s-1/q B r h s dµ 1/q .
Thus, we set

A R := C 1 (NR) β NB h s dµ 1/s-1/q
(2.10) and (2.8) is proved. Observing that if B r = NB we have equalities with constant 1 in the inequalities above, the constant C 1 works for (2.9).

Step 2. Local setup. For ℓ ∈ N, fix r 0 and ρ 0 real numbers satisfying

R ≤ r 0 < ρ 0 ≤ NR with N ℓ (ρ 0 -r 0 ) = R. For x ∈ B(x 0 , r 0 ), we have that N k N ℓ (ρ 0 -r 0 ) = N k R for k ≥ 0 so B(x, N k (ρ 0 -r 0 )) ⊂ B(x 0 , N k+1 R) ⊂ B(x, N k+ℓ+ j (ρ 0 -r 0 )),
where j = 2 when N ≥ 2 and j = ⌈(log 2 N) -1 + 1⌉ when 1 < N < 2, and consequently for any positive µ-measurable function v,

B(x,N k (ρ 0 -r 0 )) v dµ = 1 µ(B(x, N k (ρ 0 -r 0 ))) B(x,N k (ρ 0 -r 0 )) v dµ ≤ µ(B(x, N k+ℓ+ j (ρ 0 -r 0 ))) µ(B(x, N k (ρ 0 -r 0 ))) 1 B(x 0 , N k+1 R) B(x 0 ,N k+1 R) v dµ ≤ C ℓ+ j N B(x 0 ,N k+1 R) v dµ, (2.11)
where we used (2.1) in the last line. The constant C N ≥ 1 only depends on the doubling constant C d and the number N.

Step 3. Beginning of the estimate. For m > 0, set u m := min{u, m}, B r 0 := B(x 0 , r 0 ) and B ρ 0 := B(x 0 , ρ 0 ). Using the Lebesgue-Stieltjes formulation of the integral we have

B r 0 u p-q m u q dµ = (p -q) m 0 λ p-q-1 u q (B r 0 ∩ {u > λ}) dλ = (p -q) λ 0 0 λ p-q-1 u q (B r 0 ∩ {u > λ}) dλ + (p -q) m λ 0 λ p-q-1 u q (B r 0 ∩ {u > λ}) dλ ≤ λ p-q 0 u q (B r 0 ) + (p -q) m λ 0 λ p-q-1 u q (B r 0 ∩ {u > λ}) dλ =: I + II, (2.12)
where u q (A) = A u q dµ for any measurable set A ⊆ X and λ 0 is a constant chosen below.

Step 4. Choice of the threshold λ 0 . We define three functions

U(x, r) := B(x,r) u dµ, F(x, r) := B(x,r) f q dµ 1/q , G(x, r) := B(x,r) g q dµ 1/q
and for λ > λ 0 , we denote the relevant level sets by

U λ := B r 0 ∩ {u > λ}, F λ := B r 0 ∩ { f > λ}, G λ := B r 0 ∩ {g > λ}.
It follows from (2.11) with k = 0 that for x ∈ B r 0 ,

U(x, ρ 0 -r 0 ) = B(x,ρ 0 -r 0 ) u dµ ≤ C ℓ+ j N α 0 a u (NB)
and one has the same observation for F with f q . For the last term, we use (2.11) in conjunction with (2.9) to obtain

B(x,ρ 0 -r 0 ) g q dµ ≤ C ℓ+ j N NB g q dµ ≤ C ℓ+ j N C q 1 (NR) βq NB h s dµ q/s ≤ C (ℓ+ j)q/s N α q/s 0 C q 1 (NR) βq a h s (NB) q/s ,
where we used C N ≥ 1 and q/s > 1. Consequently, we choose

λ 0 := C ℓ+ j N α 0 a u (NB) + C ℓ+ j N α 0 a f q (NB) 1/q + C 1 (NR) β C ℓ+ j N α 0 a h s (NB) 1/s .
Finally, set

Ω λ := x ∈ U λ ∪ F λ ∪ G λ :
x is a Lebesgue point for u, f q and g q .

Step 5. Estimate of the measure of U λ . We begin to estimate II in (2.12) so we assume λ > λ 0 . For x ∈ B r 0 ,

(2.13) U(x, ρ 0 -r 0 ) + F(x, ρ 0 -r 0 ) + G(x, ρ 0 -r 0 ) ≤ λ 0 < λ.
On the other hand, by definition of

U λ , F λ and G λ , if x ∈ Ω λ then lim r→0 U(x, r) + F(x, r) + G(x, r) > λ
Thus for x ∈ Ω λ we can define the stopping time radius

r x := sup N -m (ρ 0 -r 0 ) : m ∈ N and U(x, N -m (ρ 0 -r 0 )) + F(x, N -m (ρ 0 -r 0 )) + G(x, N -m (ρ 0 -r 0 )) > λ .
We remark that (2.13) implies that r x < ρ 0r 0 . Of course Ω λ ⊂ ∪ x∈Ω λ B(x, r x /5).

By the Vitali Covering Lemma (5r-Covering Lemma) there exists a countable collection of balls {B(

x i , r i )} = {B i } with r i = r x i such that { 1 5 B i } are pairwise disjoint and Ω λ ⊂ ∪ i B i . Let m i ≥ 1 such that N m i r i = ρ 0 -r 0 .
We make three observations:

(i) For each i, either B i u dµ > λ 3 , ( B i f q dµ) 1/q > λ 3 , or ( B i g q dµ) 1/q > λ 3 . (ii) The radius of each B i is less than ρ 0r 0 and

x i ∈ B r 0 so B i ⊂ B ρ 0 . (iii) For 0 ≤ k < m i , N k r i = N -(m i -k) (ρ 0 -r 0 ) < ρ 0 -r 0 , so N k r i is 'above' or at
the stopping time and

N k B i u dµ + N k B i f q dµ 1/q + N k B i g q dµ 1/q ≤ Cλ,
where C shows up since we have used doubling once in the case k = 0.

Using that µ((

U λ ∪ F λ ∪ G λ ) \ Ω λ ) = 0, Ω λ ⊂ ∪ i B i and (2.4), we obtain u q (U λ ) ≤ u q (U λ ∪ F λ ∪ G λ ) ≤ i u q (B i ) ≤ i µ(B i )[Aa u (B i ) + (a f q (B i )) 1/q + r β i (a h s (B i )) 1/s ] q .
(2.14)

Then using m i ≥ 1, k α k = α and observation (iii) we obtain

a f q (B i ) = ∞ k=0 α k N k B i f q dµ = m i -1 k=0 α k N k B i f q dµ + ∞ k=m i α k N k B i f q dµ ≤ C q αλ q + ∞ k=0 α k+m i B(x i ,N k (ρ 0 -r 0 )) f q dµ, (2.15)
where we simply re-indexed the second sum and used that N m i r i = ρ 0r 0 . Now we use (2.11) and that α k+m i ≤ α k to deduce

a f q (B i ) ≤ C q αλ q + C ℓ+ j N ∞ k=0 α k N k+1 B f q dµ ≤ C q αλ q + C ℓ+ j N a f q (NB) ≤ C q αλ q + α 0 λ q 0 ≤ Cλ q ,
where we used the definition of λ 0 in Step 4 and λ > λ 0 . Similarly, a u (B i ) ≤ Cλ.

For h s , using m i ≥ 1, we obtain

(r i ) βs a h s (B i ) = ∞ k=0 α k (r i ) βs N k B i h s dµ = m i -1 k=0 α k (r i ) βs N k B i h s dµ + ∞ k=m i α k (r i ) βs N k B i h s dµ ≤ m i -1 k=0 α k N k B i g q dµ s/q + (r i ) βs ∞ k=0 α k+m i B(x i ,N k (ρ 0 -r 0 )) h s dµ,
where we used (2.8) and N k B i ⊂ NB when k < m i for the first sum, re-indexed the second sum and used that N m i r i = ρ 0r 0 . With k α k = α and observation (iii) for the first sum and α k+m i ≤ α k along with (2.11) for the second one, we deduce

(r i ) βs a h s (B i ) ≤ C s αλ s + C ℓ+ j N (NR) βs ∞ k=0 α k N k+1 B h s dµ ≤ C s αλ s + C ℓ+ j N (NR) βs a h s (NB) ≤ C s αλ s + α 0 λ s 0 C -s 1 ≤ Cλ s ,
where we used λ > λ 0 . Combining the above estimates with (2.14) we obtain

(2.16) u q (U λ ) ≤ Cλ q i µ(B i ) ≤ CC 3 d λ q i µ 1 5 B i ≤ CC 3 d λ q µ (∪ i B i )
where we used that { 1 5 B i } are pairwise disjoint. Now we use (i) and (ii) to conclude that (2.17)

∪ i B i ⊂ {M(u½ B ρ 0 ) > λ/3}∪{M( f q ½ B ρ 0 ) > (λ/3) q }∪{M(g q ½ B ρ 0 ) > (λ/3) q },
where M is the uncentered maximal function.

Step 6. Estimate of II and I. Plugging (2.16) and (2.17) into II we obtain

II = (p -q) m λ 0 λ p-q-1 u q (U λ ) dλ ≤ C(p -q) m 0 λ p-1 µ({M(u½ B ρ 0 ) > λ/3}) dλ + C(p -q) m 0 λ p-1 µ({M( f q ½ B ρ 0 ) > (λ/3) q }) dλ + C(p -q) m 0 λ p-1 µ({M(g q ½ B ρ 0 ) > (λ/3) q }) dλ =: II 1 + II 2 + II 3 .
We handle II 2 and II 3 in the same way. Using the Hardy-Littlewood maximal theorem for spaces of homogeneous type to the effect that for p ∈ (q, 2q) the L p/q → L p/q operator norm of the maximal function is bounded by C p/q (p/q)-1 , see Theorem 3.13 in [START_REF] Björn | Nonlinear potential theory on metric spaces[END_REF], we obtain

II 3 = C(p -q) m 0 λ p-1 µ({M(g q ½ B ρ 0 ) > (λ/3) q )}) dλ ≤ C p -q p X (M(g q ½ B ρ 0 )) p/q dµ ≤ C p p -q p/q-1 NB g p dµ,
where we used B ρ 0 ⊂ NB in the last step. Similarly we have that

II 2 ≤ C p p -q p/q-1 NB f p dµ.
To handle II 1 we notice that

{M(u½ B ρ 0 ) > λ/3} ⊂ {M(u½ B ρ 0 ∩{u>λ/6} ) > λ/6}.
From this estimate and the weak type (1,1) bound for the Hardy-Littlewood maximal function for spaces of homogeneous type, see Lemma 3.12 in [START_REF] Björn | Nonlinear potential theory on metric spaces[END_REF], we have

µ({M(u½ B ρ 0 ) > λ/3}) ≤ C λ B ρ 0 ∩{u>λ/6} u dµ.
Using this bound in II 1 yields

II 1 ≤ C(p -q) m 0 λ p-2 B ρ 0 ∩{u>λ/6} u dµ dλ = C(p -q) B ρ 0 u max{m,6u} 0 λ p-2 dλ dµ = C6 p-1 p -q p -1 B ρ 0 u p-1 m/6 u dµ ≤ C6 p-1 p -q p -1 B ρ 0 u p-q m u q dµ, (2.18)
and we note that we can make the constant in front of the integral arbitrarily small by choice of p > q. Combining our estimates for II 1 , II 2 and II 3 we obtain for any p ∈ (q, 2q), (2.19)

II ≤ ǫ p B ρ 0 u p-q m u q dµ + Cǫ -1 p NB f p dµ + Cǫ -1 p NB g p dµ,
where ǫ p := C(pq). Now we bound I. Note that B ⊂ B r 0 ⊂ NB. By definition of λ 0 and using (2.4),

I ≤ λ p-q 0 u q (NB) ≤ λ p-q 0 µ(NB) Aa u (NB) + (a f q (NB)) 1/q + (NR) β (a h s (NB)) 1/s q ≤ C( C ℓ+ j N ) p-q µ(NB)α p (NB), (2.20)
where we denoted α p (NB) := a u (NB) + (a f q (NB)) 1/q + (NR) β (a h s (NB)) 1/s p and put C N := max(C N , C 1/s N ) ≥ 1 on recalling that we allow for s < 1.

Step 7. Conclusion. Setting

ϕ(t) := B(x 0 ,t)
u p-q m u q dµ and combining estimates (2.12), (2. [START_REF] Macías | Lipschitz functions on spaces of homogeneous type[END_REF]) and (2.20), we may summarize our estimates as

ϕ(r 0 ) ≤ Cµ(NB)α p (NB) C (p-q)(ℓ+ j) N + ǫ p ϕ(ρ 0 ) + Cǫ -1 p NB f p dµ + Cǫ -1 p NB g p dµ,
whenever R ≤ r 0 < ρ 0 ≤ NR and N ℓ (ρ 0r 0 ) = R and where j depends at most on N, see Step 2. For notational convenience set

M 1 := Cµ(NB)α p (NB), M 2 := C NB f p dµ + C NB g p dµ, so that ϕ(r 0 ) ≤ M 1 C (p-q)ℓ N + ǫ p ϕ(ρ 0 ) + ǫ -1 p M 2 . (2.21)
Now, we set up an iteration scheme to conclude: We fix K ∈ N large enough to guarantee ∞ ℓ=0 N -Kℓ ≤ N, initiate with t 0 := R and put

t ℓ+1 := t ℓ + N -K(ℓ+1) R for ℓ = 0, 1, . . . . Then R ≤ t ℓ < t ℓ+1 ≤ NR and N K(ℓ+1) (t ℓ+1 -t ℓ ) = R so that ϕ(t ℓ ) ≤ M 1 C (p-q)K(ℓ+1) N + ǫ -1 p M 2 + ǫ p ϕ(t ℓ+1 ). (2.22)
Iterating the above inequality we obtain for any

ℓ 0 ∈ N ϕ(t 0 ) ≤ M 1 ℓ 0 ℓ=1 C (p-q)Kℓ N • ǫ ℓ-1 p + M 2 ℓ 0 ℓ=1 ǫ ℓ-2 p + ǫ ℓ 0 p ϕ(t ℓ 0 ) ≤ C M 1 + C M 2 + Cǫ ℓ 0 p ϕ(NR) provided that ǫ p ≤ (2 C (p-q)K N
) -1 ≤ 1/2 by now fixing p ∈ (q, 2q) with pq small enough.

Noting that ϕ(NR) < ∞ (by truncation of u) and t 0 = R, we may let ℓ 0 → ∞ above to conclude

ϕ(R) ≤ C M 1 + C M 2 .
Upon replacing M 1 , M 2 , α p (NB) and ϕ(R) we obtain B u p-q m u q dµ ≤ Cµ(NB) a u (NB) + (a f q (NB)) 1/q + (NR) β (a h s (NB)) (2.23)

Dividing both sides of the inequality by µ(B), taking p-th roots and letting m → ∞ we obtain (2.5) except for the presence of the term ( NB g p dµ) 1/p . We handle this term using the definition of g in terms of h and the definition of A R in (2.10), obtaining

NB g p dµ 1/p = A R NB h ps/q dµ 1/p = C 1 (NR) β NB h s dµ 1/s-1/q NB h ps/q dµ 1/p ≤ C 1 (NR) β NB h ps/q dµ q/ps-1/p NB h ps/q dµ 1/p = C 1 (NR) β NB h ps/q dµ q/sp .

Quasi-metric spaces

In Theorem 2.2, it is possible to relax the structural assumption of (X, d, µ) being a metric space by allowing the constant K in the quasi triangle inequality to take values greater than one. The proof of Theorem 2.2 does not carry over as such (or rather it becomes very technical) but we can take advantage of the fact that every quasi-metric (K > 1) is equivalent to a power of a proper metric (K = 1). See [START_REF] Aimar | On the Macías-Segovia metrization of quasi-metric spaces[END_REF][START_REF] Macías | Lipschitz functions on spaces of homogeneous type[END_REF][START_REF] Paluszyński | On quasi-metric and metric spaces[END_REF]. The following proposition is from [START_REF] Paluszyński | On quasi-metric and metric spaces[END_REF].

Proposition 3.1. Let (X, ρ) be a quasi-metric space and let 0 < δ ≤ 1 be given by (2K) δ = 2. Then there is another quasi-metric ρ such that ρδ is a metric and for all x, y ∈ X,

E -1 ρ(x, y) ≤ ρ(x, y) ≤ Eρ(x, y),
where E ≥ 1 is a constant only depending on the quasi triangle inequality constant of ρ.

With Proposition 3.1 at hand, the following theorem is a straightforward consequence of its metric counterpart. For the reader's convenience and since akin reductions to the metric case will be used at other occasions in this paper, we present the full details here. Theorem 3.2. Let (X, ρ, µ) be a space of homogeneous type. Let s, β > 0 and q > 1 be such that s < q and β ≥ D(1/s -1/q) where D is any number satisfying (2.1). Let N > 1 and (α k ) k≥0 be a non-increasing sequence of positive numbers with α :

= k α k < ∞. Define a u (B) := ∞ k=0 α k N k B u dµ
for u ≥ 0 locally integrable and B a quasi-metric ball.

Suppose that u, f, h ≥ 0 with u q , f q , h s ∈ L 1 loc (X, dµ) and there exists a constant A such that for every ball B = B(x, R),

(3.3) B u q dµ 1/q ≤ Aa u (B) + (a f q (B)) 1/q + R β (a h s (B)) 1/s .
Then there exists p > q depending on α 0 , α, A, q, s, K, N and C d such that for all balls B, B u p dµ

1/p ãu (NB) + (ã f q (NB)) 1/q + R β (ã h s (NB)) 1/s + NE 2/δ B f p dµ 1/p + R β NE 2/δ B h ps/q dµ q/sp . (3.4)
Here, ãu is obtained from a u by replacing the sequence α k with α max(0,k-j 0 -j 1 ) , where j 0 and j 1 are the minimal integers with E 2 ≤ N j 0 and E 2/δ ≤ N j 1 and E, δ are the constants from Proposition 3.1. The implicit constant depends on α 0 , α, A, q, s, β, K, N and C d .

Remark 3.5. The same remarks as after Theorem 2.2 apply.

Proof. Let d be the metric so that d 1/δ with δ ∈ (0, 1] is equivalent to ρ, provided by Proposition 3.1. Then there is a constant E > 1 only depending on the quasi-metric constant K of ρ so that

B ρ (x, r) = {z : ρ(z, x) < r} ⊆ {z : E -1 d 1/δ (z, x) < r} = B d (x, (Er) δ ) and B ρ (x, r) = {z : ρ(z, x) < r} ⊇ {z : Ed 1/δ (z, x) < r} = B d (x, (E -1 r) δ ). In total B d (x, (E -1 r) δ ) ⊆ B ρ (x, r) ⊆ B d (x, (Er) δ ).
Consequently µ is doubling with respect to the metric d, and we also see that the hypothesis (3.3) implies that

B d (x,(E -1 R) δ ) u q dµ 1/q ∞ k=0 α k N δk E δ B d (x,R δ ) u dµ + ∞ k=0 α k N δk E δ B d (x,R δ ) f q dµ 1/q + R β ∞ k=0 α k N δk E δ B d (x,R δ ) h s dµ 1/s
holds for all x ∈ X and R > 0. Setting R ′ := (E -1 R) δ , we can rewrite this as

B d (x,R ′ ) u q dµ 1/q ∞ k=0 α k N δk B d (x,E 2δ R ′ ) u dµ + ∞ k=0 α k N δk B d (x,E 2δ R ′ ) f q dµ 1/q + (R ′ ) β/δ ∞ k=0 α k N δk B d (x,E 2δ R ′ ) h s dµ 1/s .
We set N ′ := N δ . Then j 0 is the smallest positive integer so that

E 2δ ≤ (N ′ ) j 0 . We note that ∞ k=0 α k (N ′ ) k B d (x,E 2δ R ′ ) u dµ ∞ k=0 α k (N ′ ) k+ j 0 B d (x,R ′ ) u dµ ≤ ∞ k=0 α ′ k (N ′ ) k B d (x,R ′ ) udµ,
where α ′ k := α max(k-j 0 ,0) . Analogous estimates hold with f q and h s in place of u. Finally, we set

β ′ := β/δ so that (2.4) is satisfied in the metric space (X, d, µ) with (α ′ k ) k , β ′ , N ′ replacing (α k ) k , β
, N there. We also have control over the homogeneous dimension of (X, d, µ). Indeed, for x ∈ X and R > r, we see that ER > E -1 r and therefore

µ(B d (x, R δ )) µ(B d (x, r δ )) ≤ µ(B ρ (x, ER)) µ(B ρ (x, E -1 r)) E 2 R r D ,
where D is a number satisfying (2.1) for (X, ρ, µ).

It follows that D ′ = Dδ -1 satisfies (2.1) for (X, d, µ). As a consequence, β ′ = β/δ ≥ D ′ (1/s -1/q), and we can apply Theorem 2.2.

We obtain

B d u p dµ 1/p ∞ k=0 α ′ k (N ′ ) k+1 B d u dµ + ∞ k=0 α ′ k (N ′ ) k+1 B d f q dµ 1/q + (R ′ ) β ′ ∞ k=0 α ′ k (N ′ ) k+1 B d h s dµ 1/s + N ′ B d f p dµ 1/p + (R ′ ) β ′ N ′ B d h ps/q dµ q/sp
for all balls B d with radius R ′ . Note that R ′ is arbitrary. Comparing the d-balls with ρ-balls once again, we see that

B ρ (x, (E -1 r) 1/δ ) ⊂ B d (x, r) ⊂ B ρ (x, (Er) 1/δ ).
Arguing as in the beginning of the proof and denoting R = (E -1 R ′ ) 1/δ , we can get back to an inequality in the quasi-metric space (X, ρ, µ): We only need to recall that N ′ = N δ , that j 1 is the smallest integer so that E 2/δ ≤ N j 1 and set

α ′′ k := α ′ max(0,k-j 1 ) = α max(0,k-j 0 -j 1 )
. This together with the doubling condition implies that (3.4) holds.

Variants

One might wonder whether one can use in the proof of Theorem 2.2 the fractional maximal operator M βs where

M β v(x) := sup B∋x r(B) β B |v|, x ∈ X, β > 0,
to control the terms stemming from h s more efficiently. (Here r(B) denotes the radius of B.) However, this operator has no boundedness property in this generality and one has to assume volume lower bound in the following sense:

(4.1) ∃Q > 0 : ∀ balls B, µ(B) r(B) Q . Lemma 4.2.
Let (X, ρ, µ) be a space of homogeneous type. Assume that the volume has a lower bound with exponent Q > 0.

Then M β is bounded from L p (X) to L p * (X) when 1 < p and 0 < β < Q/p with p * = pQ Q-βp . For p = 1, it is weak type (1, 1 * ).
Proof. See e.g. Section 2 in [START_REF] Heikkinen | Mapping properties of the discrete fractional maximal operator in metric measure spaces[END_REF] for a simple proof on metric spaces with doubling measure that applies verbatim in the quasi-metric setting. In fact, the result follows from the inequality

M β v(x) Mv(x) 1-β/Q v β/Q 1
using the lower bound, the uncentered maximal function M and interpolation.

We obtain the following variant in the presence of a volume lower bound. Theorem 4.3. Let (X, ρ, µ) be a space of homogeneous type having a volume lower bound with exponent Q. Let s > 0, β ≥ 0 and q > 1 be such that s < q and β ≤ Q(1/s -1/q). Suppose that u, f, h ≥ 0 with u q , f q , h s ∈ L 1 loc (X, dµ) and (3.3) holds. Then there exists p > q such that (3.4) holds with R β ( NE 2/δ B h ps/q dµ) q/sp replaced by µ(B) β/Q ( NE 2/δ B h p * dµ) 1/p * where p * = pQ Q+βp .

Proof. It suffices to give a proof for (X, ρ, µ) a metric space with doubling measure. Then we can apply the general reduction argument from the previous section. In this regard, we note that if ρ is equivalent to d 1/δ , then d has lower volume bound with exponent

Q ′ := Q/δ.
For any p > q set σ := pQ s(Q+βp) = p * s . Note the condition βqs ≤ Q(qs) ensures for all p > q the bound βps < Q(ps). Hence σ > 1. Now we indicate the changes in the proof of Theorem 2.2.

One does not introduce the function g in Step 1 and the function G in Step 4 becomes H(x, t) = r β B(x,t) h s dµ 1/s . The choice of λ 0 is similar and then we can follow the argument until we need to estimate II 3 in Step 6. Here we now have

II 3 = C(p -q) m 0 λ p-1 µ({M βs (h s ½ B ρ 0 ) > (λ/3) s }) dλ X (M βs (h s ½ B ρ 0 )) p/s dµ NB h sσ dµ p/(sσ) = NB h p * dµ p/p *
by definition of σ and we used the L σ (X) → L p/s (X) boundedness of M βs from Lemma 4.2. Recall near then end of Theorem 2.2 we divide by µ(B) then take p-th roots. Thus, the power of µ(B) in front of ( NB h p * dµ) 1/p * comes from the equality

µ(B) -1 B h p * dµ p p * = µ(B) βp/Q B h p * dµ p p * . Remark 4.4. Assume β = 1, s = 2n
n+2 and q = 2 in the Euclidean space R n with Lebesgue measure, which is typical of elliptic equations. Then the Lebesgue exponent for h in Theorem 3.2 is ps q = pn n+2 while above we get p * = pn n+p , which is smaller. If β = 0, then p * = p. Of course the interest is to have β as large as possible so that p * is as small as possible, but in applications to PDEs the value of β is usually not free to choose but determined by scaling arguments. Finally note that the admissible ranges for β in the two theorems are almost complementary in this example: Indeed, since

D = Q = n, we have β ≥ n(1/s -1/q) in Theorem 2.2 and β ≤ n(1/s -1/q) in Theorem 3.2.
Another variant is to replace powers of the radius by powers of the volume already in the assumption and then no further hypothesis on the measure is required. Theorem 4.5. Let (X, ρ, µ) be a space of homogeneous type. Let s > 0, γ ≥ 0 and q > 1 be such that s < q and γ ≤ 1/s-1/q. Suppose that u, f, h ≥ 0 with u q , f q , h s ∈ L 1 loc (X, dµ) and (3.3) holds with R β (a h s (B)) 1/s replaced by µ(B) γ (a h s (B)) 1/s . Then there exists p > q such that (3.4) holds with R β ( NE 2/δ B h ps/q dµ) q/sp replaced with µ(B) γ ( NE 2/δB h sσ dµ) 1/sσ , where sσ = p 1+γp . Remark 4.6. Note that one can take γ = 0 in which case sσ = p. In accordance with Remark 4.4 we note that the higher γ the smaller the integrability needed on h.

Proof. Once again it suffices to treat the metric case. The modification to the proof of Theorem 2.2 are the same as in the above argument, except for now using instead of M βs the modified fractional maximal operator Mγs , 0 ≤ γ < 1, where [START_REF] Heikkinen | Mapping properties of the discrete fractional maximal operator in metric measure spaces[END_REF].

Mγs v(x) := sup B∋x (µ(B)) γs B |v|, x ∈ X. It maps L σ (X) into L σ 1-γsσ (X) when 1 < σ and γsσ < 1, see Remark 2.4 in

Global integrability

A typical application of Gehring's lemma is to prove higher integrability locally and globally. To extract a conclusion at the level of global spaces L p (X), we need some further hypotheses. We say that the space of homogeneous type (X, ρ, µ) is φ-regular if it satisfies φ(r) ∼ µ(B(x, r))

for all x ∈ X and r > 0, where φ : (0, ∞) → (0, ∞) is a non-decreasing function with φ(r) > 0 and φ(2r) ∼ φ(r) for r > 0. An important subclass of such spaces are the Ahlfors-David regular metric spaces where φ(r) = r Q for some Q > 0. The case of local and global different dimensions which occur on connected nilpotent Lie groups (see [START_REF] Varopoulos | Analysis and geometry on groups[END_REF]) is also covered with φ(r) ∼ r d for r ≤ 1 and φ(r) ∼ r D for r ≥ 1.

Theorem 5.1. Let (X, ρ, µ) be a φ-regular space of homogeneous type. In addition to assumptions of Theorem 3.2, suppose that u q , f q , h s ∈ L 1 (X, dµ). Then

u L p (X) u L q (X) + f L p (X) + h L ps/q (X)
with the implicit constant depending on u, f, h only through the parameters quantified in the assumption.

Proof. For the sake of simplicity let us assume N = 2 in the statement of Theorem 3.2. We shall see in Section 6.2 below that upon changing the sequence (α k ) k we can do so without loss of generality. Alternatively, we could also adapt the following argument to cover the general case. Take any R > 0 and choose a maximal R separated set of points {x i }, that is, ρ(x i , x j ) ≥ R for all i j and for every y ∈ X there exists x i such that ρ(y, x i ) < R. Since we assume that X is doubling, such a collection necessarily has only finitely many members in any fixed ball, hence, it is countable. The balls B i := B(x i , R) cover X, and there is C only depending on K and

C d such that i ½ B i (x) ≤ C
for every x ∈ X. Also the balls (2K) -1 B i are disjoint. Further, we have

(5.2) i ½ 2 k B i (x) ∼ φ(2 k R) φ(R)
for every x ∈ X and every integer k ≥ 1. Indeed, fix k ≥ 1 and x ∈ X. We can assume that 2 k-1 ≥ K, since otherwise we can just use that the left-and right-hand sides are comparable to constants depending only on K, C d and φ. Let I x be the set of i giving a non zero contribution, and N x be the cardinal of I x , that is, the value of the sum. Clearly, N x is not exceeding the number of i for which ρ(x,

x i ) ≤ 2 k R. As the balls (2K) -1 B i , i ∈ I x , are disjoint and contained in B(x, K(2 k + (2K) -1 )R), we have φ(R/2K)N x i∈I x µ((2K) -1 B i ) µ(B(x, K(2 k + (2K) -1 )R)) φ(K2 k+1 R). Also the balls B i , i ∈ I x , cover B(x, (K -1 2 k -1)R), hence φ(R/2K)N x i∈I x µ((2K) -1 B i ) i∈I x µ(B i ) µ(B(x, (K -1 2 k -1)R)) φ((K -1 2 k -1)R)≥ φ(K -1 2 k-1 R)
by the assumption on k. The claim follows using the comparability φ(

K2 k+1 R) ∼ φ(K -1 2 k-1 R) ∼ φ(2 k R) and φ(R/2K) ∼ φ(R).
Applying Theorem 3.2 in each B i and denoting by αk := α max(0,k-j 0 -j 1 ) the summable sequence appearing in the conclusion of that theorem, we see that

φ(R) -1 X u p dµ i B i u p dµ i ãu (2B i ) p + ã f p (2B i ) + R β ãh ps/q (2B i ) q/s =: I + II + III,
where we used φ-regularity, Hölder's inequality ã f q (2B i ) 1/q ã f p (2B i ) 1/p and absorbed the term with f p in (2.5) in ã f p (2B i ) and similarly for the terms with h. Let us treat III: From the continuous embedding ℓ 1 (N) ⊂ ℓ q/s (N) and (5.2) we obtain

i ãh ps/q (2B i ) q/s ≤ i ∞ k=0 αk 2 k+1 B i h ps/q dµ q/s ≤ i ∞ k=0 αk 2 k+1 B i h ps/q dµ q/s ≤ ∞ k=0 αk i 2 k+1 B i h ps/q dµ q/s ∞ k=0 αk φ(2 k+1 R) -1 X i ½ 2 k+1 B i h ps/q dµ q/s αφ(R) -1 X h ps/q dµ q/s
, where α := k αk . Doing the same for I and II, implies

(5.3) u L p (X) φ(R) 1/p-1/q u L q (X) + f L p (X) + R β φ(R) (1/p)-(q/sp) h L ps/q (X) .
Note that since R is fixed (R = 1 for example), this concludes the proof.

Remark 5.4. We note that the implicit constant in (5.3) does not depend on R. If h = 0, then we may let R → ∞ as p > q and obtain u L p (X) f L p (X) .

We have a global analogue of Theorem 4.3, which might be of independent interest as it can be proved directly in the quasi-metric setting without recursing to Section 3.

Theorem 5.5. Let (X, d, µ) be a space of homogeneous type having volume lower bound with exponent Q. Let s > 0, β ≥ 0 and q > 1 be such that s < q and β ≤ Q(1/s -1/q). Suppose that u q , f q , h s ∈ L 1 (X, dµ) and that (3.3) holds. Then there exists p > q such that u

L p (X) f L p (X) + h L p * (X) ,
where p * = pQ Q+βp . The implicit constant depends on u, f, h only through the parameters quantified in the assumption.

As for the choice of β, the same comments as in Remark 4.4 apply.

Proof. We indicate the modification to the argument of Theorem 2.2, which, as said, works directly in the quasi-metric setting for this result. This is basically the one in [START_REF] Auscher | On regularity of weak solutions to linear parabolic systems with measurable coefficients[END_REF].

There is no need for the first and second steps and the proof begins as in Step 3 without the balls B r 0 and B ρ 0 and we have X u p-q m u q dµ = (pq) m 0 λ p-q-1 u q ({u > λ}) dλ.

There is also no need for a threshold λ 0 and we set for x ∈ X and r > 0, U(x, r) := a u (B(x, r)), F(x, r) := (a f q (B(x, r)) 1/q , H(x, r) := r β (a h s (B(x, r))) 1/s , and for λ > 0, we denote U λ := {u > λ}. Note that without loss of generality we may assume α 0 ≥ 1 right from the start as this only increases the right-hand side of our hypothesis (3.3). Thus,

lim inf r→0 U(x, r) + F(x, r) + H(x, r) ≥ u(x)
for almost every x because already the first term in a u (B(x, r)) tends to α 0 u(x) ≥ u(x). We define Ũλ as the subset of U λ where this holds. Note that lim r→∞ U(x, r) + F(x, r) + H(x, r) = 0 for all x using the global assumptions on u, f, h. For the term with h, this follows from H(x, r) s r βs-Q X h s dµ, provided βs < Q, which holds under our assumption. For x ∈ Ũλ , we can define the stopping time radius

r x := sup{r > 0 : U(x, r) + F(x, r) + H(x, r) > λ}.
Remark that sup x∈ Ũλ r x < ∞. Indeed, at r = r x , U(x, r) + F(x, r) + H(x, r) = λ and therefore either U(x, r) ≥ λ/3 or F(x, r) ≥ λ/3 or H(x, r) ≥ λ/3. In the last case, we obtain r Q-βs (λ/3) s X h s dµ < ∞. The other cases also give us a bound on r. By the Vitali covering lemma, there exists a countable collection of balls {B(x i , r x i )} = {B i } such that 1 V B i are pairwise disjoint and Ũλ ⊂ ∪ i B i . (Usually, V = 5 but our metric is only a quasi-metric in which case the Vitali covering lemma still holds but with a larger constant V depending on K, and we apply it to the covering Ũλ ⊂ ∪ x∈ Ũλ B(x, V -1 r x ). A direct way to see this is by the technique in Section 3.) Now, using the hypothesis for each B i and pairwise disjointness of the balls 1 V B i ,

u q ( Ũλ ) ≤ i u q (B i ) ≤ i µ(B i ) Aa u (B i ) + (a f q (B i )) 1/q + r β i (a h s (B i )) 1/s q = i µ(B i )λ q V D i µ( 1 V B i )λ q V D µ (∪ i B i ) λ q ,
where D is the homogeneous dimension. The stopping time implies

(5.6) ∪ i B i ⊂ {Mu ≥ λ/3} ∪ {M( f q ) ≥ (λ/3) q } ∪ {M βs (h s ) ≥ (λ/3) s }.
From there the estimates are as in Step 6 and we use Lemma 4.2 for boundedness of M βs . We obtain (5.7)

X u p-q m u q dµ ≤ C(p -q) X u p-1 m u dµ + C p X f p dµ + C p X h p * dµ p/p * ,
with p * as in the statement. As X u p-1 m u dµ ≤ X u p-q m u q dµ we can hide this term if pq > 0 is small enough and then let m → ∞.

Self-improvement of the right-hand side

We discuss here the change in the exponents in the tails on the right-hand side and subsequently the change of the dilation parameter N. Both induce change in the sequence α. These remarks can be used to reduce some seemingly different properties to cases covered by Theorem 3.2.

6.1. Exponent. It is a direct consequence of the log-convexity of the L p norms that if

B u p dµ 1/p B u q dµ 1/q
with p > q, then for every s ∈ (0, q) we can write 1/q = θ/p + (1 -θ)/s for some θ ∈ (0, 1) and consequently

u L p (B,ν) u L q (B,ν) ≤ u (1-θ) L s (B,ν) u θ L p (B,ν) so that u L p (B,ν) u L s (B,ν) .
Here ν = dµ/µ(B). The same self-improving property holds true for the weak reverse Hölder inequality [16] and even for the reverse Hölder inequality with tails as we now show. To prove the claim for the inequality with tails, we use a modification of the argument from [START_REF] Bernicot | Gaussian heat kernel bounds through elliptic Moser iteration[END_REF], Appendix B. Proposition 6.1. Let (X, ρ, µ) be a space of homogeneous type. Let q ∈ (0, p), s 0 , s 1 , s 2 ∈ (0, q] with f s 1 , h s 2 ∈ L 1 loc (X) and set τ := min s 0 q , s 1 q , s 2 q . Let (α k ) k≥0 be a summable sequence of strictly positive numbers. Let N > 1 and β ≥ 0. Let ( αk ) k≥0 , (α ♯ k ) k≥0 be summable sequences of non negative numbers with α0 , α ♯ 0 > 0 and assume

(6.2) m k=0 αk α τ m-k αm , m k=0 α ♯ k α m-k α ♯ m and (6.3)
m k=0 αk α s 2 /q m-k N (m-k)βs 2 αm .
Define a u (B), ãu (B), a ♯ u (B) as in (2.3) in terms of the three respective sequences, for u ≥ 0 locally integrable, N > 1 and B a quasi-metric ball.

Assume that (6.4)

B u p dµ 1/p (a u q (B)) 1/q + b(B),
where the implicit constant does not depend on B, with

b(B) = (a f s 1 (B)) 1/s 1 + r(B) β (a h s 2 (B)) 1/s 2 .
Then, for any ball B for which a ♯ u q (B) < ∞, one has (6.5)

B u p dµ 1/p (ã u s 0 (B)) 1/s 0 + b(B),
where b is obtained by replacing α by α in the definition of b. Remark 6.6. Note that, in contrast with the improvement of integrability, we do not need the non-increasing assumption on the sequence α for this proposition. The condition (6.2) together with α ♯ 0 > 0 implies α m α ♯ m and similarly, since τ ≤ 1, we have α m αm . Hence we have to assume more than a u q (B) < ∞. For example, with τ as above, if α k = N -γk for γ > 0, then αk = N -γ ′ τk and α ♯ k = N -γ ′ k work in the theorem for any 0 < γ ′ < γ such that βs 2 < γs 2 /qγ ′ τ. In particular, decay γ > βq is needed to obtain any improvement, which typically is hard to obtain in applications. On the other hand, if β = 0, we can improve the right-hand exponent by only paying an arbitrarily small amount of decay to replace γ by γ ′ < γ. The condition (6.3) takes into account the presence of r(B) β in (6.4). Finally, the strict positivity of α k rules out in particular the case where the α k form a finite sequence, but in that case, the argument in [START_REF] Bernicot | Gaussian heat kernel bounds through elliptic Moser iteration[END_REF] already covers the situation.

Proof. Define K(δ, s 0 ) := sup (a u q (B)) 1/q (ã u s 0 (B)) 1/s 0 + b(B) + δ(a ♯ u q (B)) 1/q
, where the supremum is taken on the set of balls B such that the denominator is finite. Indeed, there is nothing to prove if the right-hand side of (6.5) is infinite, which is equivalent to the denominator being infinite since we assume a ♯ u q (B) < ∞. As α m α ♯ m , we have a u q (B) a ♯ u q (B) and the presence of δ > 0 guarantees that K(δ, s 0 ) δ -1 . We show a uniform bound in terms of δ. To this end we can of course assume K(δ, s 0 ) ≥ 1 since otherwise there is nothing to prove.

Fix a ball B with the above restriction. Let θ ∈ (0, 1) be such that

1 q = θ s 0 + 1 -θ p .
We see that

B u q dµ 1/q ≤ B u s 0 dµ θ/s 0 B u p dµ (1-θ)/p . Using (6.4), b(B) b(B) and K(δ, s 0 ) ≥ 1, B u q dµ 1/q B u s 0 dµ θ/s 0 (a u q (B)) 1/q + b(B) 1-θ B u s 0 dµ θ/s 0 K(δ, s 0 ) 1-θ (ã u s 0 (B)) 1/s 0 + b(B) + δ(a ♯ u q (B)) 1/q 1-θ K(δ, s 0 ) 1-θ (ã u s 0 (B)) 1/s 0 + b(B) + δ(a ♯ u q (B)) 1/q .
We apply this last inequality to N k B. This is possible provided N k B belongs to the same set of balls and this follows from the assumption on the sequences: For example, using (6.2),

a ♯ u (N k B) = ∞ j=k α ♯ j-k N j B u q dµ α -1 k ∞ j=k α ♯ j N j B u q dµ ≤ α -1 k a ♯ u q (B) < ∞.
Similar calculations can be done for the other terms. Thus,

a u q (B) 1/q = ∞ k=0 α k N k B u q dµ 1/q K(δ, s 0 ) (1-θ) ∞ k=0 α k ãu s 0 (N k B) q/s 0 1/q + K(δ, s 0 ) (1-θ) ∞ k=0 α k b(N k B) q 1/q + K(δ, s 0 ) (1-θ) δ ∞ k=0 α k a ♯ u q (N k B) 1/q =: I + II + III.
Each of the three sums is estimated similarly so we restrict our attention to the first one. Using the continuous embedding ℓ 1 (N) ⊂ ℓ q/s 0 (N) and the properties of α in (6.2), we compute

I s 0 ≤ ∞ k=0 α s 0 /q k ãu s 0 (N k B) = ∞ m=0 m k=0 α s 0 /q k αm-k N m B u s 0 dµ ∞ m=0 αm N m B u s 0 dµ = ãu s 0 (B).
The same kind of argument applies to the remaining two terms so that (6.7) a u q (B) 1/q K(δ, s 0 ) 1-θ (ã u s 0 (B)) 1/s 0 + b(B) + δ(a ♯ u q (B)) 1/q . We remark that it is the part of b involving r(B) β that requires us to use the strong condition (6.3). As the right-hand side is finite, we readily obtain K(δ, s 0 ) K(δ, s 0 ) 1-θ , therefore K(δ, s 0 )

1. Now, all the bounds are independent of δ, so we may send δ → 0 in (6.7). Plugging this inequality into (6.4) concludes the proof of (6.5). 6.2. Dilation. Another direction to which the reverse Hölder inequalities selfimprove is the dilation parameter on the right-hand side of

B u p dµ 1/p NB u q dµ 1/q .
Indeed, if such an inequality holds in a space of homogeneous type, then the similar inequality B u p dµ 1/p CB u q dµ 1/q holds for all balls with any C > K where K is the quasi-metric constant. See for instance Theorem 3.15 in [START_REF] Anderson | Weak A ∞ weights and weak reverse Hölder property in a space of homogeneous type[END_REF]. The proof of this fact is based on a covering of B by small balls whose N-dilates are still contained in CB and applying the weak reverse Hölder inequality in each small ball individually.

It is worth a remark that a change of geometry similar to the property just described can be carried out with the reverse Hölder inequality with tails. To formulate this technical remark, we introduce some notation. Given a sequence of positive numbers α = (α k ) k≥0 and real numbers with 1 < n ≤ m, we define the (m, n)-stretch S m,n α by (S m,n α) j := α k , j ≥ 0 : m k-1 < n j ≤ m k .

We also define the (m, n)-regrouping by (R m,n α) k :=

j:m k-1 <n j ≤m k α j + β k , k ≥ 0,
where β k is a correction term. It makes each (R m,n α) k to be the sum of equally many terms and hence the regrouping of a non-increasing sequence remains nonincreasing: The intervals

(k -1) ln m ln n , k ln m ln n contain ℓ or ℓ + 1 integers when ℓ is the integer such that ℓ < ln m ln n ≤ ℓ + 1.
We set

β k = 0 if j:m k-1 <n j ≤m k 1 = ℓ + 1 and β k = α min{ j:n j >m k-1 } otherwise.
For example, for γ > 0, the (m, n)-stretch of (m -γk ) is (term-wise) comparable to (n -γk ), and the (m, n)-regrouping of (n -γk ) is (term-wise) comparable to (m -γk ). More generally, if α is summable, so are its stretch and regrouping. For the latter, it is obvious and for the former, is follows from bounding the number of possible repetitions by 1 + ln m ln n . In addition, if α is non-increasing so are its (m, n)-stretch (m, n)-regrouping. Proposition 6.8. Let (X, ρ, µ) be a space of homogeneous type and let (α k ) k≥0 be a summable sequence of positive numbers. For u ∈ L 1 loc (X), u ≥ 0, N > 1, define

a u (B) := ∞ k=0 α k N k B u dµ.
Then for any M > 1, one has

a u (B) ∞ k=0 β k M k B u dµ, with, if M > N, β = R M,N α and if M < N, β = S M ℓ ,M R M ℓ ,N α
where ℓ is the least integer to satisfy ℓ ≥ ln N/ ln M.

Proof. We start with M > N. Then,

∞ k=0 α k N k B u dµ = ∞ k=0 j:M k-1 <N j ≤M k α j N j B u dµ ∞ k=0 j:M k-1 <N j ≤M k α j M k B u dµ ≤ ∞ k=0 (R M,N α) k M k B u dµ
as claimed, using the doubling condition.

Then there exists p > q such that for all balls B with 12B ⊂ Ω,

B u p dµ 1/p m Ω u (B) + (m Ω f q (B)) 1/q + R β (m Ω h s (B)) 1/s + 2B f p dµ 1/p + R β 2B h ps/q dµ q/sp , (7.3)
where both p and the implicit constant depend on A, D, s, q, β.

Proof. We prove the claim for B = B(x 0 , R) with x 0 ∈ X, R > 0 and 12B ⊂ Ω. We point out the relevant changes to the proof of Theorem 2.2. Having fixed B, we repeat Step 1 as before (we take N = 2) to define g q = A q R h s ½ 2B with A R a constant so that for any ball B r contained in 2B we have

r β B r h s dµ 1/s B g q dµ 1/q R β 2B h s 1/s .
In Step 2, fix r 0 and ρ 0 real numbers satisfying R ≤ r 0 < ρ 0 ≤ 2R. For x ∈ B r 0 := B(x 0 , r 0 ), we have that

B(x, ρ 0 -r 0 ) ⊂ B(x 0 , 2R) ⊂ B(x, 4R),
and consequently for any positive function v,

B(x,(ρ 0 -r 0 )) v dµ ≤ µ(B(x 0 , 2R)) µ(B(x, ρ 0 -r 0 )) B(x 0 ,2R) v dµ R ρ 0 -r 0 D B(x 0 ,2R) v dµ, (7.4)
where we used the constant D from the doubling dimension in the last line. Set γ := (R/(ρ 0r 0 )) D .

We repeat

Step 3 as it is. In Step 4, we define three functions

U(x, r) := B(x,r) u dµ, F(x, r) := B(x,r) f q dµ 1/q , G(x, r) := B(x,r) g q dµ 1/q
, and for λ > λ 0 , we denote the relevant level sets by

U λ := B r 0 ∩ {u > λ}, F λ := B r 0 ∩ { f > λ}, G λ := B r 0 ∩ {g > λ}.
We set

λ 0 := Cγm Ω u (2B) + C γm Ω f q (2B) 1/q + C(2R) β γm Ω h s (2B) 1/s ,
where C is a constant independent of u and the ball B, chosen such that, by an inclusion relation as in (7.4) we obtain

(7.5) U(x, ρ 0 -r 0 ) + F(x, ρ 0 -r 0 ) + G(x, ρ 0 -r 0 ) ≤ λ 0
for all x ∈ B(x 0 , ρ 0 ). Finally, we define as before

Ω λ := x ∈ U λ ∪ F λ ∪ G λ :
x is a Lebesgue point for u, f q and g q .

In Step 5, we note, as before, that if x ∈ Ω λ then lim r→0 U(x, r) + F(x, r) + G(x, r) > λ, and thus for x ∈ Ω λ we can define the stopping time radius, this time continuously, as

r x := sup r < ρ 0 -r 0 : U(x, r) + F(x, r) + G(x, r) > λ .
Remark that (7.5) implies that r x < ρ 0 -r 0 . Of course Ω λ ⊂ ∪ x∈Ω λ B(x, r x /5). By the Vitali Covering Lemma (5r-Covering Lemma) there exists a countable collection of balls {B(x i , r i )} = {B i } with r i = r x i such that { 1 5 B i } are pairwise disjoint and

G λ ⊂ ∪ i B i .
We make three observations: (i) For each i, either B i u dµ ≥ λ 3 , ( B i f q dµ) 1/q ≥ λ 3 , or ( B i g q dµ) 1/q ≥ λ 3 . (ii) The radius of each B i is less than ρ 0 -r 0 and x i ∈ B(x 0 , r 0 ) so B i ⊂ B(x 0 , ρ 0 ). (iii) Each r ∈ [r i , ρ 0r 0 ) is 'above' or at the stopping time and

B(x i ,r) u dµ + B(x i ,r) f q dµ 1/q + B(x i ,r) g q dµ 1/q λ.
We obtain from (7.2) that

u q (U λ ) ≤ u q (U λ ∪ F λ ∪ G λ ) ≤ i u q (B i ) i µ(B i ) m Ω,loc u (B i ) + (m Ω,loc f q (B i )) 1/q + r β i (m Ω,loc h s (B i )) 1/s q .
We handle the term involving f , to begin we split m Ω,loc f q (B i ) as

m Ω,loc f q (B i ) = sup r∈[r i , 1 2 dist (x i ,Ω c )) B(x i ,r) f q dµ = max sup r∈[r i ,ρ 0 -r 0 ) B(x i ,r) f q dµ, sup r∈[ρ 0 -r 0 , 1 2 dist (x i ,Ω c )) B(x i ,r) f q dµ . (7.6)
By observation (iii), we see that sup r∈[r i ,ρ 0 -r 0 ) B(x i ,r)

f q dµ λ.

On the other hand, sup

r∈[ρ 0 -r 0 , 1 2 dist (x i ,Ω c )) B(x i ,r) f q dµ = sup k∈[1, 1 2(ρ 0 -r 0 ) dist (x i ,Ω c )) B(x i ,k(ρ 0 -r 0 )) f q dµ sup k∈[1, 1 2(ρ 0 -r 0 ) dist (x i ,Ω c )) k(ρ 0 -r 0 ) + r 0 k(ρ 0 -r 0 ) D B(x 0 ,k(ρ 0 -r 0 )+r 0 ) f q dµ R ρ 0 -r 0 D m Ω f q (B) λ q 0 < λ q ,
where the last line is justified as follows: By the upper bound on k in the supremum, we always have

R ≤ k(ρ 0 -r 0 ) + r 0 ≤ 1 2 dist (x i , Ω c ) + r 0 ≤ 1 2 dist (x 0 , Ω c ) + 3R < 3 4 dist (x 0 , Ω c ),
where we used |x i -x| < r 0 < ρ 0 ≤ 2R and B(x 0 , 12R) ⊂ Ω. Hence every B(x 0 , k(ρ 0r 0 ) + r 0 ) is admissible in the definition of m Ω f q (B) and we get the bound claimed before since m Ω f q (B) ≤ 2 D m Ω f q (2B). Altogether, (m Ω,loc f q (B i )) 1/q λ. Terms with u and h are estimated similarly. The rest of Step 5 follows as before and we obtain

∪ i B i ⊂ {M(u½ B ρ 0 ) > λ/3} ∪ {M( f q ½ B ρ 0 ) > (λ/3) q } ∪ {M(g q ½ B ρ 0 ) > (λ/3) q }.
Step 6 involving maximal function arguments to estimate the measure of the set in the above display for λ > λ 0 as well as the overall contribution for λ < λ 0 is repeated without changes. In the end, we reach an inequality of the form (2.21). Indeed, set

ϕ(t) := B(x 0 ,t) u p-q m u q dµ, α p := m Ω u (2B) + (m Ω f q (2B)) 1/q + (R) β (m Ω h s (2B)) 1/s p
and for p ∈ (q, 2q) we may summarize our estimates as

ϕ(r 0 ) µ(B) R ρ 0 -r 0 η α p + ǫ p ϕ(ρ 0 ) + ε -1 p 2B f p dµ + ε -1 p 2B g p dµ,
whenever R ≤ r 0 < ρ 0 ≤ 2R. Here ǫ p = pq and η > 0 is independent of u and B. The claim (7.3) then follows from a well known iteration argument (see e.g. Lemma 6.1 in [12]) or from modifying the argument in Step 7.

Note that the proof for the maximal-function-like object m Ω,loc u is actually simpler than for the tailed a u . Several choices of how to discretize the scale parameters can be omitted. This setup is also very close but not comparable to Gehring's original assumption (Mu q ) 1/q Mu where q > 1 and M the Hardy-Littlewood maximal operator. Indeed, the left-hand side here does not have a maximal function and the right-hand side is a maximal function restricted to large scales (a non-local maximal function). where as before (α k ) k is a non-increasing and summable sequence of positive numbers. Then we can localize the assumptions of Theorem 2.2 to Ω. Corollary 7.7. Let Ω ⊂ X be an open set in a metric space (X, d, µ) with doubling measure. Let s, β > 0 and q > 1 be such that s < q and β ≥ D(1/s -1/q) where D is any number satisfying (2.1). Suppose that u, f, h ≥ 0 with u q , f q , h s ∈ L 1 loc (Ω, dµ) and A ≥ 0 is a constant such that for every ball B = B(x, R) with 16B ⊂ Ω (7.8)

B u q dµ 1/q ≤ Aa Ω,loc u (B) + (a Ω,loc f q (B)) 1/q + R β (a Ω,loc h s (B)) 1/s .
Then there exists p > q such that for all balls B with 32B ⊂ Ω,

B u p dµ 1/p a Ω u (4B) + (a Ω f q (4B)) 1/q + R β (a Ω h s (4B)) 1/s + 4B f p dµ 1/p + R β 4B h ps/q dµ q/sp , (7.9)
where both p and the implicit constant depend on A, β, s, q, D.

Proof. Theorem 2.2 shows how to deal with the tail. Corollary 7.1 shows how to adapt the proof to the setting relative to Ω. The proof of this Corollary can be reconstructed following the proof of Theorem 2.2 and carefully adapting the estimation in (2.15) in the spirit of estimating (7.6) to make sure that all relevant balls appearing in the estimates are contained in Ω.

7.4.

Convolutions. In the Euclidean setting where (X, d, µ) is R n equipped with the usual distance and the Lebesgue measure, we can realize the functionals a u as convolutions a u (B(x, r)) = (ϕ r * u)(x) where ϕ has suitable decay and integrability and ϕ r (x) = r -n ϕ(x/r). More precisely, our assumptions correspond to ϕ being bounded, radial, decreasing and globally integrable. A convolution makes sense in certain groups, so this kind of special functional can also be considered, for instance, in nilpotent Lie groups as in [START_REF] Varopoulos | Analysis and geometry on groups[END_REF].

Very weak A ∞ weights

For a weight (that is, a non-negative locally integrable function), the condition

B M(½ B w) dµ ≤ C B w dµ
valid for some C < ∞ and all balls B of X can be taken as a definition of the A ∞ class, where M is the uncentered maximal operator, see [8,28] for the Euclidean case with Lebesgue measure. In spaces of homogeneous type, this condition implies higher integrability with an exponent that can be computed from the constant C and the structural constants of X, see [START_REF] Hytönen | Sharp reverse Hölder property for A ∞ weights on spaces of homogeneous type[END_REF]. This was extended in [START_REF] Anderson | Weak A ∞ weights and weak reverse Hölder property in a space of homogeneous type[END_REF] to weights in the weak A ∞ class defined by (8.1)

B M(½ B w) dµ ≤ C σB w dµ.
where σ > 1 is given. The classes are shown to be independent of σ provided σ > K, K being the quasi-metric constant, and their elements still have a higher integrability. The methods passing through a dyadic analog yield an accurate estimate of the exponent in terms of the best C in the definition. We note that the dilation parameter σ is uniform: it is the same for all balls. Our methods allow us to remove the uniformity, that is we define the very weak A ∞ class as the set of weights such that for all balls, (8.2)

B M(½ B w) dµ ≤ C sup σ≥1 σB w dµ < ∞.
The quantity in the middle is the same functional as the one defined in Section 7.2 when Ω = X.

We denote by A vw ∞ this class. As the right-hand side of (8.2) requires boundedness of all averages on large balls, this rules out weights growing at ∞. For this reason, it is neither contained in, nor containing the class A weak ∞ introduced in [START_REF] Anderson | Weak A ∞ weights and weak reverse Hölder property in a space of homogeneous type[END_REF]. Typically, such very weak A ∞ weights arise from fractional equations. See the next section.

Theorem 8.3. For any very weak A ∞ weight w, there exist p > 1 and C ′ < ∞ such that for all balls B, (

B M(½ B w) p dµ 1/p ≤ C ′ sup σ≥1 σB w dµ. 8.4)
Remark 8.5. The improvement of integrability on a given ball B only depends on the finiteness of the right-hand side for that same ball and nothing else, as the proof will show. Hence, one can also define the very weak A ∞ class on B by the condition (8.2) on that very ball. The theorem remains valid if one replaces (in the assumption and the conclusion) the supremum by a tail as before. That variant leads to the class C p (see Section 8.1). The advantage is to allow some possible growth for which the tail is finite while the supremum is not. Finally our argument works with the supremum replaced by one average with a fixed dilation parameter. We leave these extensions to the interested reader. They will not be needed here.

Proof. To simplify we do the proof in the metric case. Again the trick to reduce the quasi-metric case to the metric case applies, see Section 3. The argument follows again that of Theorem 2.2 with f, h = 0 but with some changes.

We pick N = 2. We ignore Step 1 and have the setup of Step 2. Having fixed the ball B = B(x 0 , R), the parameters ρ 0 , r 0 , and ℓ such that 2 ℓ (ρ 0r 0 ) = R, define

Mv(x) := sup k∈Z B(x,2 k (ρ 0 -r 0 )) |v| dµ.
Then if M c designates the centered maximal operator,

Mv ≤ M c v ≤ Mv ≤ κ ′ M c v ≤ κ Mv. Indeed, Mv ≤ κ ′ M c v is classical, while M c v
Mv follows from the doubling property and κ does not depend on ρ 0r 0 in particular. By the same token, in the right-hand side of (8.2) we may restrict to the supremum over all σ = 2 k for integers k ≥ 0. This only causes a change in the constant C.

We modify

Step 3 as follows. With the truncation of the maximal function at level m,

B r 0 (M(½ B r 0 w)) p m dµ ≤ κ p B r 0 ( M(½ B r 0 w)) p-1 m/κ M(½ B r 0 w) dµ ≤ κ p B r 0 ( M(½ B ρ 0 w)) p-1 m/κ M(½ B ρ 0 w) dµ = κ p (p -1) m/κ 0 λ p-2 u(B r 0 ∩ {u > λ}) dλ (8.6)
with u := M(½ B ρ 0 w).

In Step 4, we pick λ 0 := C ℓ+2 d sup σ≥1 σB w dµ (which is assumed finite otherwise there was nothing to prove), where we recall that C d is the doubling constant. We observe that for x ∈ B r 0 = B(x 0 , r 0 ) and k ≥ 0,

B(x,2 k (ρ 0 -r 0 )) ½ B ρ 0 w dµ ≤ C ℓ+2 d B(x 0 ,2 k+1 R) w dµ ≤ λ 0 . (8.7) The stopping time of Step 5 is slightly different. Let λ > λ 0 . Pick x ∈ B r 0 ∩ {u > λ}. As u(x) = M(½ B ρ 0 w)(x) > λ > λ 0 , the observation above and B(x, 2 k (ρ 0 -r 0 )) ⊂ B ρ 0 when k < 0 imply u(x) = sup k<0 B(x,2 k (ρ 0 -r 0 )) ½ B ρ 0 w dµ = sup k<0 B(x,2 k (ρ 0 -r 0 )) w dµ.
Let k x < 0 be the supremum of those k < 0 for which B(x,2 k (ρ 0 -r 0 )) w dµ > λ. We extract the covering B i = B(x i , 2 k x i (ρ 0r 0 )) of B r 0 ∩ {u > λ}, where all B i are subballs of B ρ 0 with the 1 5 B i pairwise disjoint. We claim that if

B * i = 2B i , then for all x ∈ B i ∩ B r 0 , we have u(x) ≤ C 2 d M(½ B * i w)(x)
. Indeed, fix x ∈ B i ∩ B r 0 and pick k ∈ Z. In the case where k ≥ 0 we have by (8.7),

B(x,2 k (ρ 0 -r 0 )) ½ B ρ 0 w dµ ≤ λ 0 < λ < B i w dµ ≤ M(½ B i w)(x).
In the case where 0 > k ≥ k x i , we have either by the stopping time or again by (8.7)

if k = -1, B(x,2 k (ρ 0 -r 0 )) ½ B ρ 0 w dµ ≤ µ(B(x i , 2 k+1 (ρ 0 -r 0 ))) µ(B(x, 2 k (ρ 0 -r 0 ))) B(x i ,2 k+1 (ρ 0 -r 0 )) w dµ ≤ C 2 d λ < C 2 d B i w dµ ≤ C 2 d M(½ B i w)(x).
In the case where k < k x i , B(x, 2 k (ρ 0r 0 )) ⊂ B * i and B(x, 2 k (ρ 0r 0 )) ⊂ B ρ 0 , hence

B(x,2 k (ρ 0 -r 0 )) ½ B ρ 0 w dµ = B(x,2 k (ρ 0 -r 0 )) w dµ ≤ M(½ B * i w)(x).
Thus, the intermediate claim is proved. Now, using this together with (8.2) and the opening remark, we obtain

u(B r 0 ∩ {u > λ}) ≤ i u(B i ∩ B r 0 ) ≤ i B i ∩B r 0 u dµ ≤ i C 2 d B i ∩B r 0 M(½ B * i w) dµ ≤ i C 2 d µ(B * i ) B * i M(½ B * i w) dµ ≤ CC 2 d i µ(B * i ) sup k≥0 2 k B * i w dµ ≤ CC 2 d i µ(B * i ) λ λ µ(∪B i ).
The next to last inequality is by definition of B * i = 2B i , hence all the averages do not exceed λ 0 < λ by (8.7), and the last inequality uses doubling and the fact that 1 5 B i are disjoint. As B i ⊂ B ρ 0 and λ < B i w dµ we have ∪B i ⊂ B ρ 0 ∩ {M(½ B ρ 0 w) > λ} and we have obtained

u(B r 0 ∩ {u > λ}) λ µ(B ρ 0 ∩ {M(½ B ρ 0 w) > λ}).
Step 6 is now done as follows by cutting the rightmost integral in (8. We recall that ℓ was defined by 2 ℓ (ρ 0r 0 ) = R. As κ > 1, we have obtained

ϕ(r 0 ) µ(B)C pℓ d (sup σ≥1 σB w dµ) p + ǫ p ϕ(ρ 0 ).
From there, we do as in Step 7 an iteration provided p -1 is small and finally let m → ∞ to deduce (8.4).

Having this theorem at hand, we can proceed as in [START_REF] Anderson | Weak A ∞ weights and weak reverse Hölder property in a space of homogeneous type[END_REF] and show the equality of the class A vw ∞ with other classes. We say that a weight is a very weak A ∞ weight, if there exist an exponent 1 < p < ∞ and a constant C such that for all balls B and Borel subsets E of B,

(8.8) 0 < inf σ≥1 w(E) w(σB) µ(σB) µ(B) ≤ C µ(E) µ(B) 1/p .
We call A vw ∞ this class. We say that a weight w is a very weak reverse Hölder weight if there exist an exponent 1 < q < ∞ and a constant C < ∞ such that for all balls B, (8.9)

B w q dµ 1/q ≤ C sup σ≥1 σB w dµ < ∞.
We call RH vw this class.

Theorem 8.10. Let w be a weight and B be a ball of X. The condition (8.2), (8.8) for some p ∈ (1, ∞) and (8.9) for some q ∈ (1, ∞) are equivalent (with different constants). In particular, we have coincidence of A

a C p (B) := 1 |B| R n M(½ B ) p w dx B w dx + 1 |B| ∞ k=1 2 k+1 B\2 k B |B| |2 k B| p w dx ∞ k=1 2 -kn(p-1) 2 k B w dx with 1 < p < ∞ in the definition of A vw
∞ in (8.4), we recover the C p condition of Muckenhoupt [START_REF] Muckenhoupt | Norm inequalities relating the Hilbert transform to the Hardy-Littlewood maximal function[END_REF] and Sawyer [START_REF] Sawyer | Norm inequalities relating singular integrals and the maximal function[END_REF]. Namely, we say that w ∈ C p if there are δ > 0 and C > 0 so that (8.11)

w(E) ≤ C |E| |B| δ R n M(½ B ) p w

Application to a fractional divergence form equation

Throughout this section let α ∈ (0, 1). We illustrate our main results by applying them to solutions u on R n , n ≥ 2, to the fractional divergence form equation of order 2α introduced by Shieh-Spector [START_REF] Shieh | On a new class of fractional partial differential equations[END_REF]. This equation is formally given by

(D α ) * (AD α u) = (D α ) * F + f, (9.1)
where D α is the Riesz fractional gradient defined for u ∈ L 2 (R n ) as the C n -valued tempered distribution with R n -valued Fourier symbol ξ/|ξ| 1-α , that is to say, for ξ ∈ R n we have

F (D α u)(ξ) = ξ |ξ| 1-α F u(ξ).
Throughout, we use the normalization

F u(ξ) := 1 (2π) n/2 R n u(x)e -iξ•x dx
for the Fourier transform. Note that for α = 1 we would recover the classical divergence form structure with D 1 = ∇ and adjoint (D 1 ) * = -div. Also note that F in (9.1) is C n -valued whereas f is scalar valued, but since there is no danger of confusion we shall not use different notation for vector valued functions. As for the coefficients, we assume that A : R n → C n×n is bounded, measurable, and that there exists λ > 0 such that for all x ∈ R n and all ξ ∈ C n ,

λ|ξ| 2 ≤ ℜ(A(x)ξ • ξ) ≤ λ -1 |ξ| 2 . (9.2)
As in [START_REF] Shieh | On a new class of fractional partial differential equations[END_REF][START_REF] Schikorra | L p theory for fractional gradient PDE with V MO coefficients[END_REF], we study (9.1) in a global variational framework using the Hilbert space H α,2 = H α,2 (R n ) that consists of all u ∈ L 2 (R n ) such that |D α u| ∈ L 2 (R n ) and that is endowed with the natural norm u → ( u 2

L 2 + D α u 2 L 2 )
L 2 * = (L 2 * ) ′ , see Theorem 1.2.4 in [1]. Definition 9.3. Let F ∈ L 2 and f ∈ L 2 * . A function u ∈ H α,2 is called weak solution to (9.1) if for all ϕ ∈ H α,2 , R n A(x)D α u(x) • D α ϕ(x) dx = R n F(x) • D α ϕ(x) + f (x)ϕ(x) dx.
Our goal is to prove that weak solutions to (9.1) exhibit locally higher integrability of their derivatives of order α. There are at least two legitimate choices for defining such fractional derivatives: One is the Riesz fractional gradient D α u, the other one the fractional Laplacian (-∆) α/2 u that we introduce for u ∈ H α,2 through the Fourier symbol |ξ| α or, equivalently, through the singular integral representation for almost every

x ∈ R n , (-∆) α/2 u(x) = c n,α R n u(x) -u(y) |x -y| n+α dy, (9.4)
where integral is understood in the principal value sense [18,[START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF]. Our main result then reads as follows.

Theorem 9.5. Let u ∈ H α,2 and p > 2. Suppose u is a weak solution to (9.1), where f ∈ L 2 * ∩ L p * loc and F ∈ L 2 ∩ L p loc . Then there exists ǫ 0 = ǫ 0 (λ, n, α, p) > 0 such that

(1) local higher integrability |(-∆) α/2 u| + |D α u| ∈ L 2+ǫ 0 loc holds, (2) global integrability f ∈ L 2 * ∩ L p * and F ∈ L 2 ∩ L p , implies global higher integrability |(-∆) α/2 u| + |D α u| ∈ L 2+ǫ 0 .
Below, we shall prove this result by first establishing new reverse Hölder inequalities with tails for solutions to (9.1) and then applying the non-local Gehring lemmas from Theorem 2.2 and Theorem 5.1. As a consequence, there are in fact quantitative bounds that substantiate assertion (1) and ( 2) above. We shall write them out in (9.14) at the end of this section.

We remark that for equations with real V MO-coefficients A it was shown in [START_REF] Schikorra | L p theory for fractional gradient PDE with V MO coefficients[END_REF] by different techniques that global p-integrability of F leads to local p-integrability of D α u for any p > 2. The local higher integrability of D α u in (1) should therefore be seen as the counterpart of that result for equations with merely measurable coefficients. Although (-∆) α/2 u is comparable to D α u only on the global level using Fourier multipliers (the Riesz transform), we obtain local self-improvement for the former quantity, too. This, as well as the global results (2), seem to be novel for fractional divergence form equations even in the setting of [START_REF] Schikorra | Nonlinear commutators for the fractional p-Laplacian and applications[END_REF]. 9.1. Reverse Hölder estimates for the Riesz fractional gradient. We begin with a reverse Hölder estimate for the Riesz fractional gradient of solutions. Surprisingly, this estimate is still local in u and F. Proposition 9.6. Let u ∈ H α,2 be a weak solution to (9.1), where f ∈ L

2 * and F ∈ L 2 . Let ρ ∈ (2 * ,1 , 2). Then for all balls B = B(x, r) ⊂ R n , B |D α u| 2 1/2 2B |D α u| ρ 1/ρ + 2B |F| 2 1/2 + r α ∞ k=0 2 -k(1-α) 2 k B | f | 2 * ,α 1/2 * ,α
, with an implicit constant depending on n, α, λ, ρ.

Remark 9.7. The proof below will yield in fact a stronger version of the above estimate, see in particular (9.10): All but the averages for k = 0, 1 of f can be taken in L 1 instead of L 2 * ,α and ρ = 2 * ,1 is admissible in all dimensions but n = 2. For the applications we shall not need such precision.

For the proof we need to recall the notions of Riesz potential and Riesz transform. The potential I s of order s ∈ (0, 1) corresponds to the Fourier symbol |ξ| -s . For g ∈ L p , p ∈ (1, n/s), we have I s g ∈ L p * s , a bounded operator I s : L p → L p * s and an absolutely convergent representation

I s g(x) = c s R n g(y) |x -y| n-s dy for almost every x ∈ R n . For v ∈ H s,2 we have (-∆) s/2 I s v = v = I s (-∆) s/2 v.
We refer to Section V.1 in [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF] for all these properties. We shall also use the well known Riesz transform R and its adjoint R * corresponding to the symbols -iξ/|ξ| and iξ ⊤ /|ξ|, respectively. For g ∈ L p and G ∈ (L p ) n , p ∈ (1, ∞), they are given for almost every x ∈ R n by the principal value integrals [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF]. For u ∈ H α,2 we can read off from the respective Fourier symbols the important relations

Rg(x) = c n R n x -y |x -y| n+1 g(y) dy and R * G(x) = c n R n x -y |x -y| n+1 • G(y) dy, (9.8) see Section III.1 in
D α u = ∇I 1-α u = R(-∆) α/2 u. (9.9)
We are now ready to give the Proof of Proposition 9.6. We let B be a ball of radius 1 and pick an adapted cut-off function ϕ ∈ C ∞ 0 (2B) satisfying 0 ≤ ϕ ≤ 1, ϕ = 1 on B and |∇ϕ| 1. We claim that it suffices to establish the bound

R n |D α u| 2 ϕ 2 ε R n |D α u| 2 ϕ 2 + ε -1 2B |D α u| ρ 2/ρ + ε -1 2B |F| 2 + ε -1 4B | f | 2 * 2/2 * + ∞ k=2 2 -k(n+1-α) 2 k B | f | 2 (9.10)
for ε > 0 and an implicit constant depending on n, α, λ. Indeed, upon choosing ε sufficiently small to absorb the first term on the right into the left-hand side and using the defining properties of ϕ, this implies the claim for balls of radius 1 even in its stronger form alluded to in Remark 9.7. In order to pass to balls of arbitrary radius r > 0, we apply the above to x → u(xr), which is a solution to an equation of the same form as (9.1) with identical ellipticity constants.

In the following, implicit constants will only depend on n, α, λ and we shall not mention this in every single step. In order to prove (9.10) we start out with ellipticity of A, see (9.2), and write

R n |D α u| 2 ϕ 2 ≤ ℜ R n AD α u • (D α u)ϕ 2 .
We introduce the potential v := I 1-α u -2B I 1-α u with zero average on 2B. Since u ∈ L 2 we have I 1-α u ∈ L 2 * 1-α ⊂ L 2 loc and hence v ∈ L 2 loc . Moreover, ∇v = D α u ∈ L 2 due to (9.9), which leads to

R n |D α u| 2 ϕ 2 ≤ ℜ R n AD α u • ∇(vϕ 2 ) -2ℜ R n AD α u • vϕ∇ϕ =: ℜ(I) + ℜ(II).
As for II, we recall boundedness of A, ϕ, ∇ϕ, apply Young's inequality with ε and then use the standard Sobolev-Poincaré inequality for functions with zero average on a ball (Corollary 8.1.4 in [START_REF] Adams | Function Spaces and Potential Theory[END_REF]) to give

|II| ε R n |D α u| 2 ϕ 2 + ε -1 2B |v| 2 ε R n |D α u| 2 ϕ 2 + ε -1 2B |∇v| ρ 2/ρ .
Since we have ∇v = D α u, this is a desirable bound in view of (9.10).

We turn to I. Since vϕ 2 ∈ H 1,2 , we can use the Fourier transform to write ∇(vϕ 2 ) = D α (-∆) (1-α)/2 (vϕ 2 ), which in turn allows us to bring into play the equation for u with (-∆) (1-α)/2 (vϕ 2 ) ∈ H α,2 as a test function:

I = R n F • ∇(vϕ 2 ) + R n f (-∆) (1-α)/2 (vϕ 2 ) =: I 1 + I 2 .
By the support properties of ϕ and the relation D α u = ∇v we have

I 1 = 2B F • (D α u)ϕ 2 + 2B 2F • vϕ∇ϕ.
A desirable bound for the first integral on the right is obtained simply from Young's inequality with ε, whereas for the second one we also invoke the Sobolev-Poincaré inequality for v as in the treatment of II above. This completes the handling I 1 and the only term remaining is I 2 .

Let us further split I 2 into a local and a global piece and as the Fourier symbol of I α is R-valued, we can rewrite

I 2 =
I 2 = - R n RI α (½ 4B f ) • ∇(vϕ 2 ).
From f ∈ L 2 * and the above-mentioned boundedness properties of R and I α we can infer RI α (½ 4B f ) ∈ L 2 with norm controlled by f L 2 * (4B) . Hence, I 2 is of the exact same nature as I 1 and we obtain a desirable bound by the same reasoning as before upon replacing F with RI α (½ 4B f ). Finally, we use the integral representation (9.4) for (-∆) (1-α)/2 (vϕ 2 )(x) to treat the global piece I 22 . Since vϕ 2 is supported in 2B, there is no issue of convergence for x ∈ (4B) c and we get

2 k B | f (x)| dx ≤ 2B |v| 2 + ∞ k=2 2 -k(n+1-α) 2 k B | f | 2 ,
where the second step follows from the elementary inequality XY ≤ X 2 + Y 2 and Hölder's inequality on X = 2B |v|. A final application of Sobolev-Poincaré inequality to v completes the proof of (9.10). 9.2. Reverse Hölder estimates incorporating the fractional Laplacian. As our next step, we look for similar reverse Hölder estimates that incorporate (-∆) α/2 u. Instead of looking separately at the fractional Laplacian, we shall incorporate this term into the estimate for the Riesz fractional gradient and obtain a reverse Hölder estimate for the sum |D α u| + |(-∆) α/2 u|.

+ 4B |F| 2 1/2 + ∞ k=0 2 -k 2 k B |D α u| + r α ∞ k=0 2 -k(1-α) 2 k B | f | 2 * ,α 1/2 * ,α
, with an implicit constant depending on n, α, λ, ρ.

This will follow at once from the preceding proposition and the following real variable lemma that has nothing to do with solutions to (9.1). It does, however, illustrate how tails naturally enter the scene when changing the quantity to be controlled via a Riesz transform and we suggest that this phenomenon is natural also for more general Calderón-Zygmund operators. and it remains to control the right-most integral. In fact, we shall establish a pointwise bound for the integrand, which will yield the conclusion since B is normalized.

For brevity set w := D α u ∈ L 2 . Given x ∈ B, there is no issue of convergence with the integral representation (9.8) for R * (½ (2B) c w)(x) and we have

R * (½ (2B) c w)(x) = c n |y|>2 (x -y) • w(y)
|x -y| n+1 dy.

We let now z ∈ B and ε ∈ (0, 1) be free parameters to be specified later on, and introduce w 0 := ½ 2B w ∈ L 1 ∩ L 2 . Since B(z, ε) ⊂ 2B, we have

2 k B v 2 + ∞ k=1 2 -k(1-α) 2 k B |F| 2 1/2 + R α ∞ k=0 2 -k(1-α)

7. 3 .

 3 Domains. We can define the tail functional a Ω,loc u restricted to an open set Ω,

  6) at λ 0 . Let ϕ(r 0 ) := B r 0 (M(½ B r 0 w)) p m dµ. Then ϕ(r 0 ) ≤ κ p λ p-1 0 u(B r 0 ) + κ p (p -1) 1 µ(B ρ 0 ∩ {M(½ B ρ 0 w) > λ}) dλ µ

½

  4B f (-∆) (1-α)/2 (vϕ 2 ) + R n ½ (4B) c f (-∆) (1-α)/2 (vϕ 2 ) =: I 21 + I 22and treat the local piece first. Since vϕ 2 ∈ H 1,2 , we can use the Fourier transform to justify(-∆) (1-α)/2 (vϕ 2 ) = -I α R * ∇(vϕ 2 )

|

  (-∆) (1-α)/2 (vϕ 2 )(x)| ≤ c n,α 2B |v(y)| |x -y| n+1-α dy.Splitting the integral in x ∈ (4B) c into dyadic annuli, we therefore obtain|I 22 | ∞ k=2 2 -k(n+1-α) 2B |v(y)| dy

Proposition 9 .

 9 11. . Let u ∈ H α,2 be a weak solution to (9.1), where f∈ L 2 * and F L 2 . Let ρ ∈ (2 * ,1 , 2). Then for all balls B = B(x, r) ⊂ R n , B (|D α u| + |(-∆) α/2 u|) ∆) α/2 u|

Lemma 9 . 2 B 2 B

 922 12. Let α ∈ (0, 1) and u ∈ H α,2 . There is a constant C = C(n, α) such that for any ball B ⊂ R n ,|D α u| .Proof. By a scaling and translation argument it suffices again to argue for B = B(0, 1). Since R * R f =f for every f ∈ L 2 , we can split the square of the left-hand side asB |(-∆) α/2 u| 2 ≤ |R * (½ 2B R(-∆) α/2 u)| 2 + |R * (½ (2B) c R(-∆) α/2 u)| 2 .Using the relation R(-∆) α/2 u = D α u from (9.9) as well as the L 2 boundedness of R * on the first integral, we find B |(-∆) α/2 u| 2 2B |D α u| 2 + B |R * (½ (2B) c D α u)| 2 (9.13)

c - 1 4 B 4 .

 144 n R * (½ (2B) c w)(x) = |y|>2 xy |x -y| n+1 -zy |z -y| n+1 • w(y) dy + |z-y|>ε (zy) • w(y) |z -y| n+1 dy -|z-y|>ε (zy) • w 0 (y) |z -y| n+1 dy. The mean value theorem allows us to control the size of the kernel in the first integral on the right by |y| -n-1 . So, letting ε → 0 we obtain from (9.8) for almost every z ∈ B the bound |R * (½ (2B) c w)(x)| |y|>2 |w(y)| |y| n+1 dy + |R * w(z)| + |R * w 0 (z)|, with an implicit constant depending on α and n. It remains to pick z ∈ B correctly. Tchebychev's inequality entails z ∈ B : |R * w(z)| ≥ |R * w| ≤ |B| 4 and likewise the weak-(1, 1) bound for R * (with constant C = C(n) say, see Theorem II.4 in [26]) guarantees z ∈ B : |R * w 0 (z)| ≥ 4C |B| R n |w 0 | ≤ |B| Hence, the set of z ∈ B violating both conditions simultaneously has measure at least |B|/2 and we pick any such z. In conclusion, we have obtained for almost every x ∈ B the pointwise bound |R * (½ (2B) c w)(At this stage we recall w 0 = ½ 2B w, w = D α u, and thus we obtain from (9.9) thatR * w = R * R(-∆) α/2 u = -(-∆) α/2 u. Splitting the integral in |y| > 2 into dyadic annuli we eventually find |R * (½ (2B) c w)(Integrating both sides in x ∈ B completes the ongoing estimate of the right-most integral in (9.13). Now, we easily obtain a ǫ 0 (λ, n, α, p) > 0 on all balls B = B(x,

  -hand side is finite due to our assumptionsF ∈ L 2 ∩ L p loc and f ∈ L 2 * ∩ L p * loc .The global integrability stated in part (2) of the theorem follows by replacing Theorem 2.2 with Theorem 5.1 in the proof above.

  vw ∞ , A vw ∞ and RH vw . Proof. Adapt the proof of Lemma 8.2 in [3] together with our Theorem 8.3 as the proper replacement for Theorem 5.6 therein. 8.1. C p weights. Let X = R n equipped with Euclidean distance and Lebesgue measure and write |E| for the Lebesgue measure of a set E. Fix a weight w. Upon replacing the supremum sup σ>1 w(σB)/|σB| by the tail functional

  dx < ∞ holds for all balls B and measurable E ⊂ B.Following the proof of Lemma 8.2 in[START_REF] Anderson | Weak A ∞ weights and weak reverse Hölder property in a space of homogeneous type[END_REF], we see that w ∈ C p if and only if there are δ ′ > 0 and C > 0 such that for all balls B, > 0 and all balls B to the list of equivalent definitions of the C p class. In conclusion, the class C p gives examples of functions satisfying a reverse Hölder inequality with tail as in Theorem 2.2. Conversely, as we prove next a reverse Hölder inequality with a tail of the form a C p (B) for fractional derivatives of solutions to fractional divergence form equations, we see that solutions produce examples of C p weights.

	w 1+δ ′	dx	1/(1+δ ′ )	≤ Ca C p (B) < ∞.
	B
	Modifying the proof of Theorem 8.3 (see Remark 8.5), one can append
	M(½ B w) dx ≤ Ca C p (B) < ∞
	B
	holding for some C

  1/2 . Since ξ/|ξ| 1-α is comparable to |ξ| α is Euclidean norm, H α,2 coincides up to equivalent norms with the Bessel potential space that is usually denoted by the same symbol. Fractional Sobolev embedding theorems give us two important indices for every p ∈ (1, ∞): The value of α in the definition of p * and p * will usually be clear from the context. Otherwise it will be given explicitly as p * ,α or p * α . In particular, H α,2 embeds continuously into

	p * :=	pn n -αp	and	p * :=	pn n + αp	,
	that satisfy 1/p

* + 1/p * = 2/p and (p * ) * = (p * ) * = p.

Acknowledgment. We thank Tuomas Hytönen for an enlightening discussion on the topics of this work that led to the results extending the A ∞ class and Carlos Pérez for pointing out the connection to the C p class. We also thank an anonymous referee for suggesting that our results should apply to the fractional divergence form equation of Shieh-Spector [25] rather than the toy model investigated in an earlier version of our manuscript.

The first and third authors were partially supported by the ANR project "Harmonic Analysis at its Boundaries", ANR-12-BS01-0013. This material is based upon work supported by National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the MSRI in Berkeley, California, during the Spring 2017 semester. The second author was supported by the NSF INSPIRE Award DMS 1344235. The third author was supported by a public grant as part of the FMJH.

Let now M < N. Assume first that there is an integer ℓ ≥ 2 such that M ℓ = N. Then we can write

In general, we can find an integer ℓ ≥ 2 so that M ℓ-1 < N ≤ M ℓ so that by the previous case (M > N)

Extensions

There are several ways to further generalize the Gehring lemma with tails that follow by the argument used in the proof of Theorem 2.2. For the sake of clear exposition, we have not included them in the main theorem, but we briefly discuss some of them in this separate section. For simplicity we work in metric spaces. We leave the adaptations to the quasi-metric setting for the interested reader. They will not be needed in the further course. 7.1. Sequences. We usually asked the sequence α k in the definition of a u to be non-increasing. Of course, this assumption can always be relaxed by asking the sequence to be non-increasing starting from a certain index k 0 and then replacing the terms α k with 0 ≤ k ≤ k 0 with α ′ k := max 0≤k≤k 0 α k . The resulting sequence with α ′ k := α k for k > k 0 is always non-increasing and summable.

Maximal function.

The functional a u can also take the form

where Ω ⊂ X is an open set. In other words, the supremum is over "large" balls B so that 2B ⊂ Ω. We also define

Let Ω ⊂ X be an open set in a metric space (X, d, µ) with doubling measure. Let s, β > 0 and q > 1 be such that s < q and β ≥ D(1/s -1/q) where D is any number satisfying (2.1). Suppose that u, f, h ≥ 0 with u q , f q , h s ∈ L 1 loc (Ω, dµ) and A ≥ 0 is a constant such that for every ball B = B(x, R) with 2B ⊂ Ω (7.2)

Proof of Proposition 9.11. Proposition 9.6 already controls the L 2 average of D α u on B by the desired right-hand side. In order to control the L 2 average of (-∆) α/2 u, we first apply Lemma 9.12 and then use Proposition 9.6 again on the ball 2B. We begin with statement (1), where the right-hand side of the equations exhibits higher local integrability. Let us take any ρ ∈ (1, 2) to which Proposition 9.11 applies, the precise value of which does neither play a role for the argument, nor the conclusion. We introduce the tail functional

for h ≥ 0 locally integrable and B ⊂ R n a ball.

We put v := |(-∆) α u| + |(-∆) α/2 u| so that Proposition 9.11 entails for every ball

with an implicit constant depending on n, α, λ, ρ. Note that here we have been very generous by using Hölder's inequality to unite all quantities containing either (-∆) α u or D α u in one single tail for v with L ρ -averages and reducing the decay of the geometric series meeting those averages. In order to bring this estimate in the form of Theorem 2.2, we introduce v := v ρ and similarly F := |F| ρ and f := | f | ρ . In terms of v, F, f the previous bound reads

Now that the exponent of v on the right-hand side is 1, the claim follows from Theorem 2.2 after checking the numerology. The parameters in that theorem are (D, β, q, s) := n, αρ, 2 ρ , 2 * ρ and so the conditions 0 < s < q, q > 1 and β ≥ D(1/s -1/q) are satisfied. (Note that in fact D(1/s -1/q) = αρ = β and that s < 1). Hence, Theorem 2.2 gives us local higher integrability for v with exponent larger than q through the quantitative bound (2.5), provided F and f are globally integrable with exponents q and s and locally integrable to some higher exponents, respectively. By definition, this precisely means F ∈ L 2 ∩ L p loc and f ∈ L 2 * ∩ L p * loc and for some p > 2, which is our assumption.

Of course we can write the resulting estimate again in terms of the original functions: With v := |D α u| + |(-∆) α/2 | we get for all sufficiently small ǫ 0 =