NON-LOCAL GEHRING LEMMAS PASCAL AUSCHER, SIMON BORTZ, MORITZ EGERT, AND OLLI SAARI A

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NON-LOCAL GEHRING LEMMAS

PASCAL AUSCHER, SIMON BORTZ, MORITZ EGERT, AND OLLI SAARI

Abstract. We prove a self-improving property for reverse H¨older inequalities with non-local right hand side. We attempt to cover all the most important situa- tions that one encounters when studying elliptic and parabolic partial differential equations as well as certain fractional equations. We also consider non-local extensions ofA∞weights. We write our results in spaces of homogeneous type.

Contents

1. Introduction 2

2. Metric spaces 3

3. Quasi-metric spaces 11

4. Variants 14

5. Global integrability 15

6. Self-improvement of the right hand side 19

7. Extensions 23

8. Very weakA∞weights 27

9. An application to fractional equations 31

10. Technical estimates 40

References 45

2010Mathematics Subject Classification. Primary: 30L99; Secondary: 34A08, 42B25.

Key words and phrases. Gehring’s lemma, (non-local) Reverse H¨older inequalities, spaces of homogeneous type, (very weak) A∞ weights, Cp weights, fractional elliptic equations, self- improvement properties.

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.

1

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1. Introduction

Gehring’s lemma [10] establishes the open-ended property of reverse H¨older classes. If

(1.1)

1

|B|

Z

B

u^qdx 1/q

. 1

|B|

Z

B

u dx withq>1 and all Euclidean ballsB⊂Rⁿ, then

1

|B|

Z

B

u^q⁺dx 1/(q+)

.q

1

|B|

Z

B

u dx

for a certain >0 and all Euclidean balls. This self-improving property has proved to be an important tool when studying elliptic [8, 11] and parabolic [12] partial differential equations as well as quasiconformal mappings [18]. 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 H¨older inequalities when the right hand side is non-local. Understanding an analogue of Gehring’s lemma in this generality turned out to be crucial in [3], where we prove H¨older continuity in time for solutions of parabolic systems. The non-local nature arises from the use of half- order 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 H¨older 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 ∞

X

k=0

2^−k 1

|2^kB|

Z

2^kB

u dx

and certain averages over additional functions f andhthat have a special meaning in applications. The main result of this paper is Theorem3.2asserting that a variant of Gehring’s lemma, and in particular the local higher integrability ofustill 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 (Sections4 and7) as well as self-improving properties for the right-hand side of the reverse H¨older inequality with tail (Section 6). We aim at covering all the aspects that usually arise from applications. We also illustrate our main result by an application to regularity of solutions of a fractional elliptic equation different from the ones treated in [5,19,25] in Section9.

The context of our work is the following. Gehring’s lemma in a metric space endowed with a doubling measure was proved in [29]. See also the book [7].

By [21], 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 H¨older inequalities, a very clean argument using self-improving properties

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ofA∞weights was used in [2] to give an intrinsically quasi-metric proof (see also the very closely related work [16]). We do not attempt to review the literature in the Euclideann-space, but we refer to the excellent survey in [17] instead. In addition, we want to point out the recent paper on Gehring’s lemma for fractional Sobolev spaces [19]. 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 [19]

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 left hand side and/or that one may work on open subsets. In this way, our methods can also be applied to obtain a generalization ofA∞ weights: In [2], a larger class ofweak A∞weights, generalizing the one considered in [9,28] was defined and their higher (than one) integrability was proved (in spaces of homogeneous type). This class of weights, larger than the usualA∞ 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 theC_pclass studied in [22,24]. Precise definitions are given in Section8.

Acknowledgment. We thank Tuomas Hyt¨onen for an enlightening discussion on the topics of this work that led to the results extending theA∞class. We also thank Carlos P´erez for pointing out the connection to theCpclass.

2. Metric spaces

Aspace of homogeneous type(X,d, µ) is a triple consisting of a setX, 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 allx,y∈X, and

(iii) d(x,z)≤K(d(x,y)+d(y,z)) for a certainK≥1 and allx,y,z∈X;

and a Borel measureµthat is doubling in the sense that 0< µ(B(x,2r))≤C_dµ(B(x,r))<∞

holds for a certain Cd 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) ametric 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 isC>0 so that for someD>0,

(2.1) µ(B(x,R))

µ(B(x,r)) ≤C R

r D

for all x ∈ X andR ≥ r > 0. We can always takeD = log₂Cd. In the following we call this number thehomogeneous dimension(although there might be smaller positive numbers Dthan log₂Cd for which this inequality holds: our proofs work

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with any suchD). For all these basic facts on analysis in metric spaces, we refer to the book [7].

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 Theorem3.2below.

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 andβ ≥ D(1/s −1/q) where D is any number satisfying(2.1). Let N >1and let(αk)k≥0be a non-increasing sequence of positive numbers withα:=P

kαk <∞, and define

(2.3) a_u(B) :=

∞

X

k=0

αk

?

N^kB

u dµ for u≥0locally integrable and B a metric ball.

Suppose that u, f,h ≥ 0with u^q,f^q,h^s ∈ L¹_loc(X,dµ)and A ≥ 0is a constant such that for every ball B=B(x,R),

(2.4)

?

B

u^qdµ 1/q

≤ Aa_u(B)+(af^q(B))^1/q+R^β(ah^s(B))^1/s.

Then there exists p>q depending onα₀, α,A,q,s,N and Cdsuch that for all balls B,

?

B

u^pdµ 1/p

.a_u(N B)+(af^q(N B))^1/q+R^β(ah^s(N B))^1/s +

?

N B

f^pdµ 1/p

+R^β

?

N B

h^ps/qdµ q/sp

, (2.5)

with implicit constant depending onα₀, α,A,q,s, β,N and Cd.

Remark 2.6. If one assumes the sequence (αk)k≥0is finite, the functional is comparable to one single average on N^k⁰B for some k₀. This gives a proof of the classical Gehring lemma with dilated balls. Note the shift from N^k⁰B toN^k⁰⁺¹B in the conclusion. But well-known additional covering arguments show that the dilation factorN^k⁰⁺¹can be changed to any number larger than 1. If one assumes (2.7) ∃C<∞:∀k≥0 αk≤Cαk+1,

then it follows that au(N B) ≤ Cau(B) for all u ≥ 0 and all ballsB. In that case, one can replace N Bby 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 ofuonBdepends only on the higher integrability of f andhon the first dilated ballN B.

Proof. We prove (2.5) for B = B(x0,R) with x₀ ∈ XandR > 0. Throughout, we reserve the symbolCfor a constant that depends at most onα₀, α,A,q,s, β,Nand Cd but that may vary from line to line.

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Step 1. Preparation. Having fixed B, we setg^q := A^q_Rh^s1N B withAR a constant so that for any ball Brwith radiusrcontained inN B, we have

(2.8) r^β

?

B_r

h^sdµ 1/s

≤

?

B_r

g^qdµ 1/q

and (2.9)

?

N B

g^qdµ 1/q

≤C₁(NR)^β

?

N B

h^sdµ 1/s

for some C₁ depending only on the doubling condition, s and q. Indeed, write B_r= B(x,r). Asx∈N B, we haveN B= B(x0,NR)⊂B(x,2NR), hence

µ(N B)

µ(Br) ≤ µ(B(x,2NR)) µ(B(x,r)) ≤C₀

2NR r

D

≤C₀2^D NR

r

β(1/s−1/q)⁻¹

whereC₀depends only on the doubling condition. Unraveling this inequality and settingC₁=(C02^D)^1/q−1/syield

r^βµ(Br)^1/q−1/s≤C₁(NR)^βµ(N B)^1/q−1/s. Hence, asq> s,

r^βµ(Br)^1/q−1/s Z

B_r

h^sdµ

1/s−1/q

≤C₁(NR)^βµ(N B)^1/q−1/s Z

N B

h^sdµ

1/s−1/q

so that r^β

?

B_r

h^sdµ 1/s

=r^β

?

B_r

h^sdµ

1/s−1/q?

B_r

h^sdµ 1/q

≤C₁(NR)^β

?

N B

h^sdµ

1/s−1/q?

B_r

h^sdµ 1/q

.

Thus, we set

AR :=C₁(NR)^β

?

N B

h^sdµ

1/s−1/q

(2.10)

and (2.8) is proved. Observing that ifB_r =N Bwe have equalities with constant 1 in the inequalities above, the constantC₁works for (2.9).

Step 2. Local setup. For ` ∈ N, fixr₀ andρ₀ real numbers satisfyingR ≤ r₀ <

ρ₀≤NRwithN^`(ρ₀−r₀)=R. Forx∈B(x0,r₀), we have thatN^kN^`(ρ₀−r₀)=N^kR fork≥0 so

B(x,N^k(ρ₀−r₀))⊂ B(x0,N^k⁺¹R)⊂B(x,N^k⁺^`⁺^j(ρ₀−r₀)),

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where j = 2 when N ≥ 2 and j = d(log₂N)⁻¹ + 1e when 1 < N < 2, and consequently for any positiveµ-measurable functionv,

?

B(x,N^k(ρ0−r₀))

v dµ= 1

µ(B(x,N^k(ρ₀−r₀))) Z

B(x,N^k(ρ0−r₀))

v dµ

≤ µ(B(x,N^k⁺^`⁺^j(ρ₀−r₀))) µ(B(x,N^k(ρ₀−r₀)))

1 B(x₀,N^k⁺¹R)

Z

B(x0,N^k⁺¹R)

v dµ

≤C^`_N⁺^j

?

B(x0,N^k⁺¹R)

v dµ, (2.11)

where we used (2.1) in the last line. The constantCN ≥ 1 only depends on the doubling constantC_dand the numberN.

Step 3. Beginning of the estimate. For m > 0, set um := min{u,m}, Br₀ :=

B(x₀,r₀) and B_ρ₀ := B(x₀, ρ₀). Using the Lebesgue-Stieltjes formulation of the integral we have

Z

B_r

0

u_m^p−qu^qdµ=(p−q) Z m

0

λ^p−q−1u^q(Br₀∩ {u> λ})dλ

=(p−q) Z λ0

0

λ^p−q−1u^q(Br0∩ {u> λ})dλ +(p−q)

Z m

λ0

λ^p−q−1u^q(Br₀∩ {u> λ})dλ

≤λ₀^p−qu^q(Br0)+(p−q) Z m

λ₀ λ^p−q−1u^q(Br0 ∩ {u> λ})dλ

=:I+II, (2.12)

whereu^q(A)=R

Au^qdµfor any measurable setA ⊆Xandλ₀is a constant chosen below.

Step 4. Choice of the thresholdλ₀. We define three functions U(x,r) :=

?

B(x,r)u dµ, F(x,r) :=

?

B(x,r) f^qdµ 1/q

, G(x,r) :=

?

B(x,r)g^qdµ 1/q

and forλ > λ₀, we denote the relevant level sets by

U_λ := B_r₀ ∩ {u> λ}, F_λ:= B_r₀ ∩ {f > λ}, G_λ:=B_r₀∩ {g> λ}.

It follows from (2.11) withk=0 that forx∈B_r₀, U(x, ρ₀−r₀)=

?

B(x,ρ0−r₀)

u dµ≤ C^`_N⁺^j α0

au(N B)

and one has the same observation forFwith f^q. For the last term, we use (2.11) in conjunction with (2.9) to obtain

?

B(x,ρ0−r₀)

g^qdµ≤C^`_N⁺^j

?

N B

g^qdµ≤C^`_N⁺^jC^q₁(NR)^βq

?

N B

h^sdµ q/s

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≤ C^(`_N⁺^j)q/s

α^q/s₀ C^q₁(NR)^βqah^s(N B)^q/s, where we usedC_N ≥1 andq/s>1. Consequently, we choose

λ₀:= C^`_N⁺^j

α₀ au(N B)+ C^`_N⁺^j

α₀ af^q(N B)

!1/q

+C₁(NR)^β C^`_N⁺^j

α₀ ah^s(N B)

!1/s

.

Finally, set Ωλ :=n

x∈U_λ∪F_λ∪G_λ :xis a Lebesgue point foru, f^qandg^q o.

Step 5. Estimate of the measure ofU_λ. We begin to estimate IIin (2.12) so we assumeλ > λ₀. Forx∈B_r₀,

(2.13) U(x, ρ₀−r₀)+F(x, ρ₀−r₀)+G(x, ρ₀−r₀)≤λ₀ < λ.

On the other hand, by definition ofU_λ,F_λandG_λ, if x∈Ωλthen limr→0U(x,r)+F(x,r)+G(x,r)> λ Thus forx∈Ωλwe can define the stopping time radius

r_x:=sup n

N^−m(ρ₀−r₀) :m∈N

andU(x,N^−m(ρ0−r₀))+F(x,N^−m(ρ0−r₀))+G(x,N^−m(ρ0−r₀))> λo . We remark that (2.13) implies thatrx < ρ₀−r₀. Of courseΩλ ⊂ ∪_x∈_Ω_λB(x,rx/5).

By the Vitali Covering Lemma (5r-Covering Lemma) there exists a countable collection of balls{B(xi,ri)} = {B_i}withri = rx_i such that{¹₅Bi}are pairwise disjoint andΩλ ⊂ ∪iBi. Letmi ≥1 such thatN^mⁱri =ρ₀−r₀.

We make three observations:

(i) For eachi, either>

B_iu dµ > ^λ₃, (>

B_i f^qdµ)^1/q> ^λ₃, or (>

B_ig^qdµ)^1/q > ^λ₃. (ii) The radius of eachBiis less thanρ₀−r₀andxi∈Br₀ soBi⊂ B_ρ₀.

(iii) For 0≤k<m_i,N^kr_i =N^−(mⁱ^−k)(ρ0−r₀)< ρ0−r₀, soN^kr_iis ‘above’ or at the stopping time and

?

N^kBi

u dµ+

?

N^kBi

f^qdµ 1/q

+

?

N^kBi

g^qdµ 1/q

≤Cλ,

whereC shows up since we have used doubling once in the casek=0.

Using thatµ((U_λ∪F_λ∪G_λ)\Ωλ)=0,Ωλ⊂ ∪iBiand (2.4), we obtain u^q(Uλ)≤u^q(Uλ∪F_λ∪G_λ)≤X

i

u^q(Bi)

≤X

i

µ(Bi)[Aau(Bi)+(af^q(Bi))^1/q+r^β_i(ah^s(Bi))^1/s]^q. (2.14)

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Then usingmi≥1,P

kαk =αand observation (iii) we obtain a_f^q(Bi)=

∞

X

k=0

αk

?

N^kB_i

f^qdµ=

m_i−1

X

k=0

αk

?

N^kB_i

f^qdµ+

∞

X

k=m_i

αk

?

N^kB_i

f^qdµ

≤C^qαλ^q+

∞

X

k=0

αk+m_i

?

B(xi,N^k(ρ₀−r₀))

f^qdµ, (2.15)

where we simply re-indexed the second sum and used thatN^mⁱri = ρ₀−r₀. Now we use (2.11) and thatαk+m_i ≤αk to deduce

a_f^q(Bi)≤C^qαλ^q+C^`_N⁺^j

∞

X

k=0

αk

?

N^k⁺¹B

f^qdµ

≤C^qαλ^q+C^`_N⁺^ja_f^q(N B)≤C^qαλ^q+α₀λ^q₀

≤Cλ^q,

where we used the definition ofλ0in Step 4 andλ > λ0. Similarly,a_u(Bi) ≤ Cλ.

Forh^s, usingmi≥1, we obtain (ri)^βsah^s(Bi)

=

∞

X

k=0

αk(ri)^βs

?

N^kBi

h^sdµ

=

mi−1

X

k=0

αk(ri)^βs

?

N^kB_i

h^sdµ+

∞

X

k=m_i

αk(ri)^βs

?

N^kB_i

h^sdµ

≤

mi−1

X

k=0

αk

?

N^kB_i

g^qdµ s/q

+(ri)^βs

∞

X

k=0

αk+m_i

?

B(xi,N^k(ρ0−r₀))

h^sdµ, where we used (2.8) andN^kB_i ⊂ N Bwhenk<m_ifor the first sum, re-indexed the second sum and used thatN^mⁱri = ρ₀−r₀. WithP

kαk = αand observation (iii) for the first sum andαk+m_i ≤αkalong with (2.11) for the second one, we deduce

(ri)^βsah^s(Bi)≤C^sαλ^s+C^`_N⁺^j(NR)^βs

∞

X

k=0

αk

?

N^k+1B

h^sdµ

≤C^sαλ^s+C^`_N⁺^j(NR)^βsah^s(N B)≤C^sαλ^s+α₀λ₀^sC^−s₁

≤Cλ^s,

where we usedλ > λ₀. Combining the above estimates with (2.14) we obtain (2.16) u^q(U_λ)≤Cλ^qX

i

µ(Bi)≤CC_d³λ^qX

i

µ ¹₅B_i

≤CC_d³λ^qµ(∪iB_i) where we used that{¹₅Bi}are pairwise disjoint. Now we use (i) and (ii) to conclude that

(2.17) ∪_iB_i ⊂ {M(u1Bρ0)> λ/3}∪{M(f^q1Bρ0)>(λ/3)^q}∪{M(g^q1Bρ0)>(λ/3)^q}, whereMis the uncentered maximal function.

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Step 6. Estimate ofIIandI. Plugging (2.16) and (2.17) intoII we obtain II=(p−q)

Z m

λ0

λ^p−q−1u^q(Uλ)dλ

≤C(p−q) Z m

0

λ^p−1µ({M(u1Bρ0)> λ/3})dλ +C(p−q)

Z m

0

λ^p−1µ({M(f^q1Bρ0)>(λ/3)^q})dλ +C(p−q)

Z m

0

λ^p−1µ({M(g^q1B_ρ

0)>(λ/3)^q})dλ

=:II₁+II₂+II₃.

We handle II₂ andII₃in the same way. Using the Hardy-Littlewood maximal theorem for spaces of homogeneous type and recalling that theL^p/q →L^p/qoper- ator norm of the maximal function is bounded byC_(p/q)−1^p/q , we obtain

II₃=C(p−q) Z m

0

λ^p−1µ({M(g^q1Bρ0)>(λ/3)^q)})dλ

≤Cp−q p

Z

X

(M(g^q1Bρ0))^p/qdµ

≤C p

p−q

p/q−1Z

N B

g^pdµ,

where we usedB_ρ₀ ⊂N Bin the last step. Similarly we have that II₂ ≤C p

p−q

p/q−1Z

N B

f^pdµ.

To handleII₁we notice that

{M(u1Bρ0)> λ/3} ⊂ {M(u1Bρ0∩{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 we have

µ({M(u1Bρ0)> λ/3})≤ C λ

Z

Bρ0∩{u>λ/6}u dµ.

Using this bound inII₁yields II₁≤C(p−q)

Z m

0

λ^p−2 Z

Bρ0∩{u>λ/6}u dµdλ

=C(p−q) Z

Bρ0

u

Z max{m,6u}

0

λ^p−2dλdµ

=C6^p−1p−q p−1

Z

Bρ0

u_m/6^p−1u dµ

≤C6^p−1p−q p−1

Z

Bρ0

u_m^p−qu^qdµ, (2.18)

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and we note that we can make the constant in front of the integral arbitrarily small by choice of p>q. Combining our estimates forII₁,II₂andII₃we obtain for any

p∈(q,2q),

(2.19) II ≤p

Z

Bρ0

u_m^p−qu^qdµ+C⁻¹_p Z

N B

f^pdµ+C⁻¹_p Z

N B

g^pdµ, wherep:=C(p−q).

Now we boundI. Note thatB⊂ Br₀ ⊂N B. By definition ofλ₀and using (2.4), I ≤λ₀^p−qu^q(N B)

≤λ₀^p−qµ(N B) Aau(N B)+(af^q(N B))^1/q+(NR)^β(ah^s(N B))^1/sq

≤C(Ce^`_N⁺^j)^p−qµ(N B)αp(N B), (2.20)

where we denotedαp(N B) := a_u(N B)+(af^q(N B))^1/q+(NR)^β(ah^s(N B))^1/sp

and putCe_N :=max(CN,C_N^1/s)≥1 on recalling that we allow fors<1.

Step 7. Conclusion. Setting ϕ(t) :=

Z

B(x0,t)u_m^p−qu^qdµ

and combining estimates (2.12), (2.19) and (2.20), we may summarize our estimates as

ϕ(r₀)≤Cµ(N B)αp(N B)Ce^(p−q)(`_N ⁺^j)+pϕ(ρ₀)+C⁻¹_p Z

N B

f^pdµ+C⁻¹_p Z

N B

g^pdµ, wheneverR≤ r₀< ρ₀≤ NRandN^`(ρ₀−r₀) =Rand where jdepends at most on N, see Step 2. For notational convenience set

M₁:=Cµ(N B)αp(N B), M₂:=C

Z

N B

f^pdµ+C Z

N B

g^pdµ, so that

ϕ(r₀)≤ M₁Ce^(p−q)`_N +pϕ(ρ₀)+⁻¹_p M₂. (2.21)

Now, we set up an iteration scheme to conclude: We fix K ∈ Nlarge enough to guaranteeP∞

`=0N^−K` ≤ N, initiate witht₀ := Rand putt_`₊₁ :=t_`+N^−K(`⁺¹⁾Rfor

`=0,1, . . .. ThenR≤t_` <t_`₊₁≤ NRandN^K(`⁺¹⁾(t`+1−t_`)=Rso that ϕ(t_`)≤M₁Ce^(p−q)K(`_N ⁺¹⁾+⁻¹_p M₂+pϕ(t_`₊₁).

(2.22)

Iterating the above inequality we obtain for any`₀∈N ϕ(t₀)≤ M₁

`0

X

`=1

Ce^(p−q)K`_N ·^`−_p ¹+M₂

`0

X

`=1

_p^`−²+^`_p⁰ϕ(t_`₀)

≤C M₁+C M₂+C^`_p⁰ϕ(NR)

provided thatp ≤ (2Ce_N^(p−q)K)⁻¹ ≤1/2 by now fixingp ∈(q,2q) with p−qsmall enough.

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Noting that ϕ(NR) < ∞(by truncation ofu) and t₀ = R, we may let `₀ → ∞ above to conclude

ϕ(R)≤C M₁+C M₂. Upon replacing M₁,M₂,αp(N B) andϕ(R) we obtain

Z

B

u_m^p−qu^qdµ≤Cµ(N B) au(N B)+(af^q(N B))^1/q+(NR)^β(ah^s(N B))^1/sp

+C Z

N B

f^pdµ+C Z

N B

g^pdµ.

(2.23)

Dividing both sides of the inequality byµ(B), takingp−th roots and lettingm→ ∞ we obtain (2.5) except for the presence of the term (>

N Bg^pdµ)^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

?

N B

g^pdµ 1/p

=AR

?

N B

h^ps/qdµ 1/p

=C₁(NR)^β

?

N B

h^sdµ

1/s−1/q?

N B

h^ps/qdµ 1/p

≤C₁(NR)^β

?

N B

h^ps/qdµ

q/ps−1/p?

N B

h^ps/qdµ 1/p

=C₁(NR)^β

?

N B

h^ps/qdµ q/sp

.

3. Quasi-metric spaces

In Theorem2.2, 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 Theorem2.2does 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 [1,21,23]. The following proposition is from [23].

Proposition 3.1. Let(X, ρ)be a quasi-metric space and let0< δ≤ 1be given by (2K)^δ=2. Then there is another quasi-metricρ˜such thatρ˜^δis a metric and for all x,y∈X,

E⁻¹ρ(x,y)≤ρ(x,˜ y)≤Eρ(x,y),

where E ≥1is a constant only depending on the quasi triangle inequality constant ofρ.

With Proposition3.1at hand, the following theorem is a straightforward consequence of its metric counterpart. For the reader’s convenience and since akin re- ductions 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 > 1be such that s <q andβ ≥ D(1/s−1/q)where D is any number satisfying

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(2.1). Let N > 1 and(αk)k≥0 be a non-increasing sequence of positive numbers withα:=P

kαk <∞. Define au(B) :=

∞

X

k=0

αk

?

N^kB

u dµ for u≥0locally integrable and B a quasi-metric ball.

Suppose that u, f,h ≥0with u^q, f^q,h^s∈ L¹_loc(X,dµ)and there exists a constant A such that for every ball B= B(x,R),

(3.3)

?

B

u^qdµ 1/q

≤ Aa_u(B)+(af^q(B))^1/q+R^β(ah^s(B))^1/s.

Then there exists p > q depending onα₀, α,A,q,s,K,N and C_d such that for all balls B,

?

B

u^pdµ 1/p

.a˜_u(N B)+(˜a_f^q(N B))^1/q+R^β(˜a_h^s(N B))^1/s +

?

NE^2/δB

f^pdµ 1/p

+R^β

?

NE^2/δB

h^ps/qdµ q/sp

. (3.4)

Here, a˜_u is obtained from a_u by replacing the sequence αk with αmax(0,k−j₀−j₁), where j₀ and j₁ are the minimal integers with E² ≤ N^j⁰ and E^2/δ ≤ N^j¹ and E, δ are the constants from Proposition 3.1. The implicit constant depends on α₀, α,A,q,s, β,K,N and Cd.

Remark3.5. The same remarks as after Theorem2.2apply.

Proof. Letdbe the metric so thatd^1/δwithδ∈(0,1] is equivalent toρ, provided by Proposition3.1. Then there is a constantE>1 only depending on the quasi-metric constantKofρso that

B^ρ(x,r)={z:ρ(z,x)<r} ⊆ {z:E⁻¹d^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⁻¹r)^δ).

In total B^d(x,(E⁻¹r)^δ) ⊆ B^ρ(x,r)⊆ B^d(x,(Er)^δ). Consequentlyµis doubling with respect to the metricd, and we also see that the hypothesis (3.3) implies that

?

B^d(x,(E⁻¹R)^δ)

u^qdµ 1/q

.

∞

X

k=0

αk

?

N^δkE^δB^d(x,R^δ)

u dµ

+

∞

X

k=0

αk

?

f^qdµ

!1/q

+R^β

∞

X

k=0

αk

?

h^sdµ

!1/s

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holds for allx∈XandR>0. SettingR⁰:=(E⁻¹R)^δ, we can rewrite this as

?

B^d(x,R⁰)

u^qdµ 1/q

.

∞

X

k=0

αk

?

N^δkB^d(x,E^2δR⁰)

u dµ

+

∞

X

k=0

αk

?

f^qdµ

!1/q

+(R⁰)^β/δ

∞

X

k=0

αk

?

h^sdµ

!1/s

.

We set N⁰ := N^δ. Then j₀is the smallest positive integer so thatE^2δ ≤ (N⁰)^j⁰. We note that

∞

X

k=0

α_k

?

(N⁰)^kB^d(x,E^2δR⁰)

u dµ.

∞

X

k=0

α_k

?

(N⁰)^k⁺^j⁰B^d(x,R⁰)

u dµ≤

∞

X

k=0

α⁰_k

?

(N⁰)^kB^d(x,R⁰)

udµ,

whereα⁰_k := α_max(k−j₀,0). Analogous estimates hold with f^q andh^sin place ofu.

Finally, we setβ⁰ :=β/δso that (2.4) is satisfied in the metric space (X,d, µ) with (α⁰_k)k, β⁰,N⁰ replacing (αk)k, β,N there. We also have control over the homogeneous dimension of (X,d, µ). Indeed, forx ∈XandR>r, we see thatER> E⁻¹r and therefore

µ(B^d(x,R^δ))

µ(B^d(x,r^δ)) ≤ µ(B^ρ(x,ER)) µ(B^ρ(x,E⁻¹r)) .

E²R r

^D , whereDis a number satisfying (2.1) for (X, ρ, µ).

It follows that D⁰ = Dδ⁻¹ satisfies (2.1) for (X,d, µ). As a consequence,β⁰ = β/δ≥ D⁰(1/s−1/q), and we can apply Theorem2.2.

We obtain

?

B^d

u^pdµ 1/p

.

∞

X

k=0

α⁰_k

?

(N⁰)^k⁺¹B^d

u dµ+

∞

X

k=0

α⁰_k

?

(N⁰)^k⁺¹B^d

f^qdµ

!1/q

+(R⁰)^β⁰

∞

X

k=0

α⁰_k

?

(N⁰)^k⁺¹B^d

h^sdµ

!1/s

+

?

N⁰B^d

f^pdµ 1/p

+(R⁰)^β⁰

?

N⁰B^d

h^ps/qdµ q/sp

for all balls B^d with radiusR⁰. Note thatR⁰ is arbitrary. Comparing thed-balls with ρ-balls once again, we see thatB^ρ(x,(E⁻¹r)^1/δ) ⊂ B^d(x,r) ⊂ B^ρ(x,(Er)^1/δ).

Arguing as in the beginning of the proof and denotingR= (E⁻¹R⁰)^1/δ, we can get back to an inequality in the quasi-metric space (X, ρ, µ): We only need to recall that N⁰ = N^δ, that j₁ is the smallest integer so that E^2/δ ≤ N^j¹ and set α⁰⁰_k := α⁰_max(0,k−_j

1) = α_max(0,k−j₀−j₁). This together with the doubling condition implies

that (3.4) holds.

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4. Variants

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

M^βv(x) :=sup

B3x

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 ofB.) However, this operator has no boundedness property in this generality and one has to assumevolume lower boundin the following sense:

(4.1) ∃Q>0 : ∀ballsB, 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^βis bounded from L^p(X)to L^p^∗(X) when1< p and0< β <Q/p with p^∗= _Q−βp^pQ . For p=1, it is weak type(1,1^∗).

Proof. See e.g. Section 2 in [15] 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^βv(x) . Mv(x)^1−β/Qkvk^β/Q₁ using the lower bound, the

uncentered maximal functionMand 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≥ 0with u^q,f^q,h^s ∈ L¹_loc(X,dµ)and(3.3) holds. Then there exists p > q such that (3.4)holds with R^β(>

NE^2/δBh^ps/qdµ)^q/sp replaced byµ(B)^β/Q(>

NE^2/δBh^p^∗dµ)^1/p^∗ where p∗= _Q^pQ₊_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 ifρis equivalent tod^1/δ, thend has lower volume bound with exponentQ⁰:=Q/δ.

For anyp>qsetσ:= _s(Q^pQ₊_βp) = ^p_s^∗. Note the conditionβqs≤Q(q−s) ensures for all p > q the boundβps < Q(p− s). Hence σ > 1. Now we indicate the changes in the proof of Theorem2.2.

One does not introduce the function gin Step 1 and the function G in Step 4 becomesH(x,t)=r^β >

B(x,t)h^sdµ1/s

. The choice ofλ₀is similar and then we can follow the argument until we need to estimateII₃in Step 6. Here we now have

II₃ =C(p−q) Z m

0

λ^p−1µ({M^βs(h^s1Bρ0)>(λ/3)^s})dλ .

Z

X

(M^βs(h^s1B_ρ

0))^p/sdµ .

Z

N B

h^sσdµ p/(sσ)

= Z

N B

h^p^∗dµ p/p∗

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by definition of σ and we used theL^σ(X) → L^p/s(X) boundedness of M^βs from Lemma4.2. Recall near then end of Theorem2.2we divide byµ(B) then takep-th roots. Thus, the power ofµ(B) in front of (>

N Bh^p^∗dµ)^1/p^∗ comes from the equality µ(B)⁻¹Z

B

h^p^∗dµ _p∗^p

=µ(B)^βp/Q

?

B

h^p^∗dµ _p∗^p

.

Remark 4.4. Assumeβ = 1, s = _n²ⁿ₊₂ andq = 2 in the Euclidean spaceRⁿ with Lebesgue measure, which is typical of elliptic equations. Then the Lebesgue exponent for hin Theorem 3.2 is ^ps_q = _n^pn₊₂ while above we get p∗ = _n^pn₊_p, which is smaller. Ifβ = 0, then p∗ = p. Of course the interest is to haveβas large as possible so that p_∗is as small as possible, but in applications to PDEs the value of βis usually not free to choose but determined by scaling arguments. Finally note that the admissible ranges forβin the two theorems are almost complementary in this example: Indeed, sinceD=Q=n, we haveβ≥ n(1/s−1/q) in Theorem2.2 andβ≤n(1/s−1/q) in Theorem3.2.

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 ≥0and q>1be such that s<q andγ ≤1/s−1/q. Suppose that u,f,h≥0with u^q,f^q,h^s∈ L¹_loc(X,dµ) and(3.3) holds with R^β(ah^s(B))^1/sreplaced byµ(B)^γ(ah^s(B))^1/s. Then there exists p >q such that(3.4)holds with R^β(>

NE^2/δBh^ps/qdµ)^q/spreplaced with µ(B)^γ(>

NE^2/δBh^sσdµ)^1/sσ, where sσ= ₁₊^p_γp.

Remark 4.6. Note that one can takeγ = 0 in which case sσ = p. In accordance with Remark4.4we note that the higherγthe smaller the integrability needed on h.

Proof. Once again it suffices to treat the metric case. The modification to the proof of Theorem2.2are the same as in the above argument, except for now using instead ofM^βsthe modified fractional maximal operator ˜M^γs, 0≤γ <1, where

M˜^γsv(x) :=sup

B3x

(µ(B))^γs

?

B

|v|, x∈X.

It mapsL^σ(X) intoL^1−γsσ^σ (X) when 1< σandγsσ <1, see Remark 2.4 in [15].

5. 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 spacesL^p(X), we need some further hypotheses. We say that the space of homogeneous type (X, ρ, µ) is φ-regularif it satisfies

φ(r)∼µ(B(x,r))

for all x ∈ Xandr > 0, where φ : (0,∞) → (0,∞) is a non-decreasing function with φ(r) > 0 andφ(2r) ∼ φ(r) forr > 0. An important subclass of such spaces are theAhlfors–David regularmetric spaces whereφ(r)=r^Qfor someQ>0. The

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case of local and global different dimensions which occur on connected nilpotent Lie groups (see [27]) is also covered withφ(r) ∼ r^d for r ≤ 1 andφ(r) ∼ r^Dfor r ≥1.

Theorem 5.1. Let(X, ρ, µ)be aφ-regular space of homogeneous type. In addition to assumptions of Theorem3.2, suppose that u^q,f^q,h^s∈L¹(X,dµ). Then

kuk_Lp(X).kuk_Lq(X)+kfk_Lp(X)+khk_Lps/q(X)

with the implicit constant depending on u,f,h only through the parameters quan- tified in the assumption.

Proof. For the sake of simplicity let us assume N = 2 in the statement of Theo- rem3.2. We shall see in Section6.2below that upon changing the sequence (αk)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 Rseparated set of points {x_i}, that is, ρ(xi,xj) ≥Rfor alli, jand for everyy∈Xthere exists xisuch thatρ(y,xi)<R.

Since we assume thatXis doubling, such a collection necessarily has only finitely many members in any fixed ball, hence, it is countable. The balls Bi := B(xi,R) coverX, and there isConly depending onKandC_d such that

X

i

1B_i(x)≤C

for everyx∈X. Also the balls (2K)⁻¹Biare disjoint. Further, we have

(5.2) X

i

12^kB_i(x)∼ φ(2^kR) φ(R)

for every x ∈ X and every integer k ≥ 1. Indeed, fixk ≥ 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 onK,Cdandφ. LetIxbe the set ofigiving a non zero contribution, andN_x be the cardinal ofI_x, that is, the value of the sum. Clearly,Nxis not exceeding the number ofifor whichρ(x,xi)≤ 2^kR.

As the balls (2K)⁻¹B_i,i∈I_x, are disjoint and contained inB(x,K(2^k +(2K)⁻¹)R), we have

φ(R/2K)Nx .X

i∈I_x

µ((2K)⁻¹Bi).µ(B(x,K(2^k+(2K)⁻¹)R)).φ(K2^k⁺¹R).

Also the ballsB_i,i∈I_x, coverB(x,(K⁻¹2^k−1)R), hence φ(R/2K)N_x&X

i∈I_x

µ((2K)⁻¹B_i)&X

i∈I_x

µ(B_i)&µ(B(x,(K⁻¹2^k−1)R))

&φ((K⁻¹2^k−1)R)≥φ(K⁻¹2^k−1R)

by the assumption onk. The claim follows using the comparabilityφ(K2^k⁺¹R) ∼ φ(K⁻¹2^k−1R)∼φ(2^kR) andφ(R/2K)∼φ(R).