Journal of Functional Analysis 263, (22) 3883 3899. SHARP REVERSE HÖLDER PROPERTY FOR A WEIGHTS ON SPACES OF HOMOGENEOUS TYPE TUOMAS HYTÖNEN, CARLOS PÉREZ, AND EZEQUIEL RELA Abstract. In this article we present a new proof of a sharp Reverse Hölder Inequality for A weights. Then we derive two applications: a precise open property of Muckenhoupt classes and, as a consequence of this last result, we obtain a simple proof of a sharp weighted bound for the Hardy-Littlewood maximal function involving A constants: ( ) /p M L p (w) c p [w]ap [σ] A, where < p <, σ = w p and c is a dimensional constant. Our approach allows us to extend the result to the context of spaces of homogeneous type and prove a weak Reverse Hölder Inequality which is still sufficient to prove the open property for A p classes and the L p boundedness of the maximal function. In this latter case, the constant c appearing in the norm inequality for the maximal function depends only on the doubling constant of the measure µ and the geometric constant κ of the quasimetric.. Introduction and main results.. Introduction. In this article we present a new proof of a sharp Reverse Hölder Inequality (RHI) for A weights. The sharpness of the result relies on the precise dependence of the exponent involved on the A constant of the weight. We improve on the result from [HP], where there is proved a sharp RHI for R d. We present here a new and simpler approach, with the extra advantage of allowing us to extend the result to any space of homogeneous type. In this case, we obtain a weak RHI which is still sharp in the dependence on the A constant of the weight. Furthermore this new approach gives a better result within the simplest context which for the sake of clarity we present first in Section 2 both the statement and the proof. The rest of the paper is devoted to the general case, namely when the underlying space is of homogeneous type. 99 Mathematics Subject Classification. Primary: 4225. Secondary: 43A85. Key words and phrases. Space of homogeneous type, Muckenhoupt weights, Reverse Hölder, Maximal functions. The first author is supported by the European Union through the ERC Starting Grant Analytic-probabilistic methods for borderline singular integrals, and by the Academy of Finland, grants 366 and 33264. The second author is supported by the Spanish Ministry of Science and Innovation grant MTM29-8934, the second and third authors are also supported by the Junta de Andalucía, grant FQM-4745.
2 TUOMAS HYTÖNEN, CARLOS PÉREZ, AND EZEQUIEL RELA We also present two results derived from the sharp RHI: first, a precise open property for A p weights (Theorem.2), which is new even in the standard case of R d for A p classes defined over cubes. y this we mean that we exhibit, for the classical A p A p ε theorem, a quantitative analysis of the dependence of ε on the A constant of the weight. This sort of result was already stated in [uc93], but here we present an improvement of that estimate. As a second consequence of our main result, we provide a simple proof of a mixed weighted bound for the Hardy-Littlewood maximal function (Theorem.3), originally proved in [HP] in the usual context of R d. There are two reasons why this new result is of interest. First because it gives a new refinement of the well-known improvement of Muckenhoupt s classical theorem due to S. M. uckley [uc93]. The second reason is because it also gives an interesting and unexpected improvement of the so-called A 2 theorem ([Hyt2] [HPTV]) as shown in [HP] (see also [HL] and [HLP]). One of the main difficulties arising in the setting of spaces of homogeneous spaces is the absence of dyadic cubes, which is a useful and commonly used tool in analysis on metric spaces. In [Chr9], Christ developed a substitute for dyadic cubes which has been exploited since then to overcome this obstacle. See, for instance, [AI5], where the authors study the relation between dyadic and classical maximal functions. There is also proved a qualitative Reverse Hölder Inequality, which yields the standard A p A p ε theorem. See the recent work [HK] for a construction of dyadic systems in this generality with application to weighted inequalities. However, we present here an approach avoiding the use of dyadic sets, and we work directly with the natural quasimetric, following carefully the dependence on the geometric constants. Some of the main ideas come from [MP98] where it was crucial to avoid the dyadic sets in order to get sharp bounds. Let us start with some standard definitions. A quasimetric d on a set S is a function d : S S [, ) which satisfies () d(x, y) = if and only if x = y; (2) d(x, y) = d(y, x) for all x, y; (3) there exists a finite constant κ such that, for all x, y, z S, d(x, y) κ(d(x, z) + d(z, y)). As usual, given x S and r >, let (x, r) = {y S : d(x, y) < r} be the ball with center x and radius r. If = (x, r) is a ball, we denote its radius r by r() and its center x by x. A space of homogeneous type (S, d, µ) is a set S together with a quasimetric d and a nonnegative orel measure µ on S such that the doubling condition (.) µ((x, 2r)) Cµ((x, r)) holds for all x S and r >. As usual, the dilation of a ball (x, λr) with λ > will be denoted by λ.
REVERSE HÖLDER INEQUALITY ON SPACES OF HOMOGENEOUS TYPE 3 If C µ is the smallest constant for which (.) holds, then the number D µ = log 2 C µ is called the doubling order of µ. y iterating (.), we have ( ) µ() (.2) µ( ) C2+log 2 µ κ r() Dµ r( ) for all balls. Note that C 2+log 2 κ µ = (4κ) Dµ. A particular case that we will use is the following elementary inequality. Let be a ball and let λ >. Then (.3) µ(λ) (2λ) Dµ µ() Throughout this paper, we will say that a constant c = c(κ, µ) > is a structural constant if it depends only on the quasimetric constant κ and the doubling constant C µ. The latter will often appear as a dependence on the doubling order D µ. In a general space of homogeneous type, the balls (x, r) are not necessarily open, but by a theorem of Macías and Segovia [MS79], there is a continuous quasimetric d which is equivalent to d (i.e., there are positive constants c and c 2 such that c d (x, y) d(x, y) c 2 d (x, y) for all x, y S) for which every ball is open. We always assume that the quasimetric d is continuous and that balls are open. We will adopt the usual notation: if ν is a measure and E is a measurable set, ν(e) denotes the ν-measure of E. Also, if f is a measurable function on (S, d, µ) and E is a measurable set, we will use the notation f(e) := f(x) dµ. We also will denote the µ-average of f over a ball as E f = fdµ = µ() fdµ. In several occasions we will need Lebesgue s differentiation theorem, so we will assume that it is valid for the spaces under study. This will follow if we assume that the family of continuous functions with compact support is a dense family in L (µ). We recall that a weight w (any nonnegative measurable function) satisfies the A p condition for < p < if [w] Ap := sup ( ) ( w dµ w p dµ) p, where the supremum is taken over all the balls in S. Since the A p classes are increasing with respect to p, we can define the A class in the natural way by A := p> A p. This class of weights can also be characterized by means of an appropriate constant. In fact, there are various different definitions of this constant, all of them equivalent in the sense that they define the same class of weights. Perhaps the more classical and known definition is the following due to Hruščev [Hru84] (see also [GCRdF85]): [w] exp A := sup ( ) ( ) w dµ exp log w dµ.
4 TUOMAS HYTÖNEN, CARLOS PÉREZ, AND EZEQUIEL RELA However, in [HP] the authors use a new A constant (which was originally introduced by Fujii in [Fuj78] and later by Wilson in [Wil87]), which seems to be better suited. Let M stand for the usual uncentered Hardy- Littlewood maximal operator: Mf(x) = sup f dµ. x Then we can define [w] A := [w] W A := sup M(wχ ) dµ. w() When the underlying space is R d, it is easy to see that [w] A c[w] exp A for some structural c >. In fact, it is shown in [HP] that there are examples showing that [w] A is much smaller than [w] exp A. The same line of ideas yields the inequality in this wider scenario. We refer the reader to the recent work of eznosova and Reznikov [R] for a more complete study of these different A constants in dimension..2. Main results. We first present the statement of our main theorem in full generality, within the context of a space of homogeneous type (S, d, µ). We have the following theorem: Theorem. (Sharp weak Reverse Hölder Inequality). Let w A. Define the exponent r(w) as r(w) = + := + τ κµ [w] A 6(32κ 2 (4κ 2 + κ) 2, ) Dµ [w] A where κ is the quasimetric constant and D µ is the doubling order of the measure from (.2). Then, ( /r(w) w dµ) r(w) 2(4κ) Dµ w dµ, 2κ where is any ball in S. Note that on the right hand side, we have a dilation of the ball, and that is the reason of calling this estimate a weak inequality. We also remark here that the fact that on the right hand side we obtain a structural constant is crucial for the applications. We refer the reader to the work of Kinnunen [Kin98] for an A version of a Reverse Hölder Inequality. As we already mentioned, from this theorem we will derive two results. The first is a precise open property for A p weights. We explicitly compute an admissible value of ε > such that any A p weight w belongs to A p ε. Theorem.2 (The Precise Open property). Let < p < and let w A p. Recall that for a weight w we defined in Theorem. the quantity r(w) =
REVERSE HÖLDER INEQUALITY ON SPACES OF HOMOGENEOUS TYPE 5 + τ κµ[w] A. Then w A p ε where ε = p r(σ) = p + τ κµ [σ] A where as usual σ = w p and p is the dual exponent of p: p + p =. Furthermore, (.4) [w] Ap ε 2 p (4κ) pdµ [w] Ap Finally, as an application of this last result, we present a short proof of the following mixed bound for the maximal function. Theorem.3. Let M be the Hardy-Littlewood maximal function and let < p < as above, and σ = w p. Then there is a structural constant A µκ such that ( ) /p (.5) M L p (w) A µκ p [w] A p [σ] A, Of course this theorem improves uckley s theorem: M L p (w) c µκ p [w] p A p. This paper is organized as follows. In Section 2 we present the proof of the RHI in the simplest case: R d with standard cubes and Lebesgue measure. Then, in Section 3 we present the proof of a weak version of the RHI for spaces of homogeneous type and then we derive the applications. We choose to present the two cases separately since in the first one the main ideas appear in a very clean way. Some of those arguments cannot be extended directly to the general case, so the proofs presented in Section 3 require an additional effort and we are able to obtain only a weak version of the RHI. Nevertheless, it will be shown that this weak version is good enough for the proof of the open property for A p weights and for the improvement of uckley s theorem with sharp constants. 2. The proof for the classical setting In this section we present the proof of the sharp RHI for R d with the Euclidean metric, Lebesgue measure and A p classes defined over cubes. The main advantage here is that we can use maximal functions adapted to those cubes. Since we will be working with dyadic children of an arbitrary cube, the appropriate definition of A constant is the following. Definition 2.. For a weight defined on R n, we define the A constant as [w] A := sup M(wχ Q ) dx. Q w(q) where the supremum is taken over all cubes with edges parallel to the coordinate axes. As usual, when this supremum is finite, we say that the weight w belongs to the A class. Q
6 TUOMAS HYTÖNEN, CARLOS PÉREZ, AND EZEQUIEL RELA We start with the following lemma. It is interesting on its own, since it can be viewed as a self-improving property of the maximal function when restricted to A weights. Lemma 2.2. Let w be any A weight and let be a cube. Then for any < ε, we have that 2 d+ [w] A ( ) +ε (2.) (M(χ Q w)) +ε dx 2[w] A w dx, where M denotes the dyadic maximal function associated to the cube. Proof. We can assume, since all calculations will be performed on, that the weight w is supported on that cube, that is, w = wχ Q. Define Ω λ := {Mw > λ}. We start with the following identity: (Mw) +ε dx = = wq ελ ε Mw(Ω λ ) dλ ελ Q ε Mw dλ + ελ ε Mw(Ω λ ) dλ w Q Now, for λ w Q, there is a family of maximal nonoverlapping dyadic cubes {Q j } j for which Ω λ = Q j and w dx > λ. j Q j Therefore, by using this decomposition and the definition of the A constant, we can write (2.2) (Mw) +ε dx wq ε [w] A w( ) + ελ ε Mw dxdλ. w Q j Q j y maximality of the cubes in {Q j } j, it follows that the dyadic maximal function M can be localized: Mw(x) = M(wχ Qj )(x), for any x Q j, for all j N. Now, if we denote by Q the dyadic parent of a given cube Q, then we have that Mw dx = M(wχ Qj ) [w] A w(q j ) [w] A w( Q j ) Q j Q j Therefore, j Q j Mw dx j = [w] A w Qj Q j [w] A λ2 d Q j [w] A λ2 d Q j [w] A λ2 d Ω λ,
REVERSE HÖLDER INEQUALITY ON SPACES OF HOMOGENEOUS TYPE 7 and then (2.2) becomes (Mw) +ε dx w ε [w] A w( ) + ε[w] A 2 d w Q λ ε Ω λ dλ. Averaging over, we obtain that (Mw) +ε dx w +ε Q [w] A + ε2d [w] A (Mw) +ε dx. + ε To conclude with the proof, we can obtain the desired inequality for any < ε 2 d+ [w] A by absorbing the last term into the left. We now have the following theorem. We remark that in this standard case, we can recover (and improve) on the known sharp RHI, with no dilations involved. Theorem 2.3 (Sharp Reverse Hölder Inequality). Let w A and let be a cube. Then ( ) +ε w +ε dx 2 w dx, for any ε > such that < ε 2 d+ [w] A. efore we proceed with the proof, we remark here that the parameter is better than the one obtained in [HP]. 2 d+ [w] A Proof. We assume again that w = wχ Q. We clearly have that w +ε dx (Mw) ε wdx. Now we argue in a similar way as in the previous lemma to obtain that (Mw) ε wdx = ελ ε w(ω λ ) dλ wq = ελ ε w( ) dλ + ελ ε w(ω λ ) dλ w Q w ε w( ) + w Q ελ ε j w(q j ) dλ, where the cubes {Q j } j are from the decomposition of Ω λ above. Therefore, (Mw) ε wdx wq ε w( ) + ε2 d λ ε Q j dλ w Q j wq ε w( ) + ε2 d λ ε Ω λ dλ w Q w ε w( ) + ε2d + ε (Mw) +ε dx.
8 TUOMAS HYTÖNEN, CARLOS PÉREZ, AND EZEQUIEL RELA Averaging over we obtain w +ε dx w +ε + ε2d + ε (Mw) +ε dx. Now we use Lemma 2.2 to conclude with the proof: w +ε dx w +ε + ε2d + ε (Mw) +ε dx ( w +ε + ε2d+ [w] A w dx + ε ( ) +ε 2 w dx ) +ε since, by hypothesis, ε2d [w] A +ε 2. 3. Proofs for the general case of spaces of homogeneous type efore we proceed with the proofs of our results, we need to introduce a local version of a Calderón-Zygmund lemma valid for spaces of homogeneous type from [MP98]. We need some notation first. Definition 3.. Let be a ball and let δ > be fixed. We use the notation = ( + δ)κ, where κ is the quasimetric constant of d. We also define the following family: (3.) =,δ = { : x and r() δr( )}. Given an integrable function f on, the maximal function of f associated to is defined by (3.2) M f(x) = sup :x f dµ if x belongs to an element of the basis, and M f(x) = otherwise. We remark here that with this definition, the maximal operator depends on the reference ball and, in addition, on the parameter δ, which can be any positive number. Therefore, if λ > and we define (3.3) Ω λ = {x S : M f(x) > λ}, then (3.4) Ω λ. An important property of the family is expressed in the following lemma. The proof follows easily from (.2). Lemma 3.2. Let be any ball and let be defined as in (3.). Let f be a locally integrable function on. Then: () Any ball is contained in.
REVERSE HÖLDER INEQUALITY ON SPACES OF HOMOGENEOUS TYPE 9 (2) If λ < f, then ( ) f /Dµ r() 2κ 2 ( + δ) r( ). λ It follows then that, for any N >, whenever f > λ and λ r() δ N r( ) ( ) 2κ 2 Dµ (+δ)n δ f. We now present the local Calderón Zygmund covering lemma in this context which is from [MP98]. Lemma 3.3 (Calderón Zygmund decomposition). Let δ > and let f be a nonnegative and integrable function on = ( + δ)κ. For N > and ) Dµ λ f, define the set Ω λ as in (3.3) If Ω λ is not empty, ( 2κ 2 (+δ)n δ then there exists a countable family { i } of pairwise disjoint balls such that i) i i Ω λ i i, where = (4κ 2 + κ). ii) r( i ) δ N r( ) for all i. iii) For all i, λ < fdµ. i iv) If η i and η 2, then f ηi λ. The next lemma contains a localization argument for the maximal function, which is a key ingredient in the proof. For the dyadic case in R d, this was a direct consequence of the maximality of the cubes in the Calderón- Zygmund decomposition. In the general setting of spaces of homogeneous type we have the following substitute. We borrow the idea from [MP98, Lemma 4.4]. Lemma 3.4. Consider, for a fixed λ as in the previous lemma, the Calderón- Zygmund decomposition of the set Ω λ with N 2κ. Define L = (8κ 2 ) Dµ. Then, for any ball i and any x i Ω Lλ, we have that (3.5) M f(x) M (fχ )(x), i where = ( ) = (4κ 2 + κ) 2. Proof. Let x i Ω Lλ. Then there exists a ball containing x such that Lλ < f dµ. Now we claim that r() r(i ). Suppose not, then r() > r( i ) and therefore the ball is contained in := (x i ; 2κr()). y the doubling
TUOMAS HYTÖNEN, CARLOS PÉREZ, AND EZEQUIEL RELA property of µ from (.2), we have that µ( ) f dµ µ() f dµ L f dµ. The ball clearly belongs to, since r( ) = 2κr() 2κδ N r( ) δr( ). In addition, under the hypothesis that r() > r(i ), we have that = 2κ r() r( i ) i = η i with η > 2. Therefore, property iv) of Lemma 3.3 implies that f λ. This implies that f Lλ, which is a contradiction. Then the claim is true and r() r(i ). It is clear that in this case the ball is contained in i and then f dµ f χ i dµ M (fχ )(x). i Now we present the proof of the Reverse Hölder inequality and its applications. We start with a preliminary lemma, which is the generalization of Lemma 2.2. Lemma 3.5. Let w be any A weight. Then there is a structural constant τ µκ such that, for any < ε τ µκδ [w] A, we have that +ε (3.6) (M w) +ε dµ 3[w] A ( w dµ). The constant τ µκδ can be taken as τ µκδ = 6(6κ 2 (4κ 2 + κ) 2 ( + δ ))Dµ, where κ is the quasimetric constant and D µ is the doubling order of the measure from (.2). Proof. As in the proof of Lemma 2.2, we can assume that the weight is localized, in this case on the ball, namely w = wχ, since by definition of the local maximal function, the values of w outside are ignored (recall (3.4)). We write (3.7) Ω λ = {x : M w(x) > λ}. ( ) For N (4κ 2 + κ) 2 and Γ = 2κ 2 (+δ)n Dµ, δ we write (M w) +ε dµ = ελ ε M w(ω λ ) dλ Γw = ελ ε M w(ω λ ) dλ + = Γ ε w ε Γw ελ ε M w(ω λ ) dλ M w dµ + ελ ε M w(ω λ ) dλ Γw
REVERSE HÖLDER INEQUALITY ON SPACES OF HOMOGENEOUS TYPE Now we use the Calderón-Zygmund decomposition from Lemma 3.3 with the choice for N above. For the first term, since we are only considering the values of w on, we can use the definition of the A constant. Then we obtain (3.8) (M w) +ε dµ Γ ε w ε [w] A w( ) + ελ ε M w dµdλ, Γw i i where the family { i } i has the properties listed in that lemma. Now we focus on a fixed i and compute the integral of the maximal function as follows. Consider L as in Lemma 3.4, L = (8κ 2 ) Dµ and the partition of i = 2 3 where = i Ω Lλ, 2 = i Ω λ \ Ω Lλ, 3 = i \ Ω λ. Then, using Lemma 3.4, we obtain M w dµ = M w dµ + M w dµ + M w dµ i 2 3 M (wχ ) dµ + Lλµ( i i ) + λµ(i ), i where we use in the last term that the inclusion i implies that M w(x) λ for any x i \ Ω λ. Now define, to abbreviate, θ = 4κ 2 + κ. Then = θ and by the doubling property (.3), we have that µ(i ) (2θ)Dµ µ( i ). Now, again by definition of [w] A, we have M w dµ [w] A w(i ) + 2Lλ(2θ) Dµ µ( i ) i ( [w] A w (2θ) 2Dµ + 2Lλ(2θ) Dµ) µ( i i ) 3L(2θ) 2Dµ [w] A λµ( i ) since, by the choice of N, the average of the weight over i is smaller than λ. Now we can continue with the sum from (3.8): w ελ ε i i M w dµdλ 3L(2θ) 2Dµ [w] A Γw 3L(2θ) 2Dµ [w] A 3εL(2θ)2Dµ [w] A + ε Γw ελ ε i µ( i ) dλ ελ ε µ(ω λ ) dλ (M w) +ε dµ
2 TUOMAS HYTÖNEN, CARLOS PÉREZ, AND EZEQUIEL RELA Finally, collecting all estimates and taking the average over, we obtain (M w) +ε dµ Γ ε w +ε [w] A + 3εL(2θ)2Dµ [w] A (M w) +ε dµ. + ε y the hypothesis on ε, we have that 3εL(2θ) 2Dµ [w] A + ε 2, and therefore the last term can be absorbed by the left hand side. In addition, one can verify (with some tedious computations) that Γ ε 3 2 for ε τ µκδ. We obtain (M w) +ε dµ 3[w] A w +ε, and the proof is complete. Now we are ready to present the new proof of the weak Reverse Hölder Inequality with sharp exponent. Proof of Theorem.. Let w be an A weight and let be a fixed ball. We remark here that the idea is to use Lemma 3.5 where we made the assumption on the localization of the weight, namely w = wχ. Note that in that lemma, the maximal operator depends on δ, and any positive δ will work. ut due to the blow-up of ε on the endpoint δ =, we will choose δ away from, namely δ =. Therefore, the hypothesis on ε that we will use are that < ε τ µκ[w] A, with τ µκ = 6(32κ 2 (4κ 2 + κ) 2 ) Dµ. With this assumption on δ, the inequality we need to prove is the following: (3.9) ( ) w +ε +ε dµ 2(4κ) D µ w dµ, for any ε > as above. As in Section 2 above, we can bound the weight by the maximal function. Then, (3.) w +ε dµ (M w) ε wdµ. For the term on the right hand side, we proceed in a similar way as in the previous lemma, considering a Calderón-Zygmund decomposition of the level set Ω λ = {x : M w(x) > λ}.
REVERSE HÖLDER INEQUALITY ON SPACES OF HOMOGENEOUS TYPE 3 with N κ(4κ + ). Then, (M w) ε wdµ = = Γw ελ ε w(ω λ ) dλ Γ ε w ε w( ) + ελ ε w( ) dλ + Γw Γw ελ ε i ελ ε w(ω λ ) dλ w( i ) dλ, where Γ = ( 4κ 2 N ) D µ and we use, as before, that i = θ i = (4κ 2 + κ) i and thus w(i ) = w i µ(i ) λ(2θ) Dµ µ( i ). Therefore, (M w) ε wdµ Γ ε w ε w( ) + ε(2θ) Dµ λ ε w( i ) dλ Γw i Γ ε w ε w( ) + ε(2θ) Dµ λ ε µ(ω λ ) dλ Γ ε w ε w( ) + ε(2θ)dµ + ε Γw (M w) +ε dµ. Then, averaging in (3.) over, we obtain µ() µ( ) w +ε dµ Γ ε w +ε + ε(2θ)dµ + ε (M w) +ε dµ. Now we note that by hypothesis we have that ε is in the range allowed in Lemma 3.5. We use again the doubling property (.3) and then we obtain the desired estimate: w +ε dµ (4κ) Dµ ( Γ ε w +ε + ε(2θ)dµ + ε ( (4κ) Dµ Γ ε + ε(2θ)dµ 3[w] A + ε ) +ε 2(4κ) ( Dµ w dµ ) (M w) +ε dµ ) ( ) +ε w dµ To check the last inequality, it is easy to verify that with this choice of ε, we also have that ε(2θ)dµ 3[w] A (+ε) 2 and, as before, Γε 3 2. This completes the proof of the weak version of the RHI stated in (3.9). 3.. Precise open property for Muckenhoupt classes. Proof of Theorem.2. Let w A p and denote, as usual, the dual weight w p = σ. We choose ε = p r(σ), which is the same as r(σ) = p p ε (observe that ε > and p ε > ). We can easily compute the following
4 TUOMAS HYTÖNEN, CARLOS PÉREZ, AND EZEQUIEL RELA ( p ε dµ) w (p ε) = ( ) p w ( p )r(σ) r(σ) dµ ( 2(4κ) Dµ ) p σ dµ by the sharp weak RHI. Now, for the A p ε constant of w, we proceed as follows. Let be any ball. Then, by the doubling property (.3) of the measure, we have that ( ) p ε ( ) p w dµ w (p ε) 2 p (4κ) pdµ w dµ σ Taking the supremum over all balls, we obtain and therefore the proof is complete. [w] Ap ε 2 p (4κ) pdµ [w] Ap, 3.2. Sharp uckley s theorem with mixed constants. Proof of Theorem.3. We start by pointing out that we have the following analogue of the standard case of R d (with cubes) for the known weak norm estimate for the maximal function: (3.) M L q, (w) (2θ) Dµ q [w] A q < q <, where θ = 4κ 2 + κ and κ is the quasimetric constant. Consider, for any nonnegative measurable function f and λ >, the level set Ω λ = {x S : Mf(x) > λ}. y a Vitali type covering lemma ([SW92], Lemma 3.3) we can obtain a countable family of balls { j } j such that µ( j ) where, as before, = θ. Therefore j f dµ > λ and Ω λ j λ q w(ω λ ) λ q w(j ) j ( ) q w(j ) fw q w q dµ µ( j j ) j (2θ) Dµq w(j ) ( ) q ( µ( j j ) µ(j ) σ dµ f q w dµ j j (2θ) Dµq [w] Aq f q L q (w) and then (3.) follows. We will also use that, for f t := fχ f>t, the following inclusion holds: j {x S : Mf(x) > 2t} {x S : Mf t (x) > t} ) q
REVERSE HÖLDER INEQUALITY ON SPACES OF HOMOGENEOUS TYPE 5 Now, we write the integral from the L p norm and compute Mf p L p (w) = p t p w{y S : Mf(y) > t} dt t = p2 p t p w{y S : Mf(y) > 2t} dt t p2 p t p w{y S : Mf t (y) > t} dt t p2 p (2θ) Dµ(p ε) [w] Ap ε t p f p ε t w dµdt tp ε t p2 2p (4κ) pdµ (2θ) Dµ(p ε) [w] Ap = p2 2p (4κ) pdµ (2θ) Dµ(p ε) [w] A p ε y the precise open property we can take ε = p choice, we finally obtain that S S S r(σ) = f(y) f p w dµ t ε dt t f p ε w dµ p +τ κµ[σ] A. With this M p L p (w) p22p (4κ) pdµ (2θ) Dµ(p ε) ( + τ κµ [σ] A )[w] Ap f p w dµ, p S and this yields (.5) and therefore the proof is complete. [AI5] [R] [uc93] [Chr9] [Fuj78] References Hugo Aimar, Ana ernardis, and ibiana Iaffei, Comparison of Hardy- Littlewood and dyadic maximal functions on spaces of homogeneous type, J. Math. Anal. Appl. 32 (25), no., 5 2. Oleksandra eznosova and Alexander Reznikov, Equivalent definitions of dyadic Muckenhoupt and reverse Hölder classes in terms of Carleson sequences, weak classes, and comparability of dyadic L log L and A constants, Preprint, arxiv:2.52 (22). Stephen M. uckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 34 (993), no., 253 272. Michael Christ, A T (b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 6/6 (99), no. 2, 6 628. Nobuhiko Fujii, Weighted bounded mean oscillation and singular integrals, Math. Japon. 22 (977/78), no. 5, 529 534. [GCRdF85] José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 6, North- Holland Publishing Co., Amsterdam, 985. [HK] [HL] [HLP] Tuomas Hytönen and Anna Kairema, Systems of dyadic cubes in a doubling metric space, Preprint, arxiv:2.985 (2). Tuomas Hytönen and Michael T. Lacey, The A p A inequality for general calderon zygmund operators, Preprint, arxiv:6.4797 (2). Tuomas Hytönen, Michael T. Lacey, and Carlos Pérez, Non-probabilistic proof of the A 2 theorem, and sharp weighted bounds for the q-variation of singular integrals, Preprint, arxiv:22.2229 (22).
6 TUOMAS HYTÖNEN, CARLOS PÉREZ, AND EZEQUIEL RELA [HP] Tuomas Hytönen and Carlos Pérez, Sharp weighted bounds involving A, Anal. PDE, (to appear). [HPTV] Tuomas Hytönen, Carlos Pérez, Sergei Treil, and Alexander Volberg, Sharp weighted estimates for dyadic shifts and the A 2 conjecture, J. Reine Angew. Math., (to appear). [Hru84] Sergei V. Hruščev, A description of weights satisfying the A condition of Muckenhoupt, Proc. Amer. Math. Soc. 9 (984), no. 2, 253 257. [Hyt2] Tuomas Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Annals of Math. 75 (22), no. 3, 476 56. [Kin98] Juha Kinnunen, A stability result on Muckenhoupt s weights, Publ. Mat. 42 (998), no., 53 63. MR 62862 (99e:4225) [MP98] Paul MacManus and Carlos Pérez, Generalized Poincaré inequalities: sharp self-improving properties, Internat. Math. Res. Notices (998), no. 2, 6. [MS79] Roberto A. Macías and Carlos Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (979), no. 3, 257 27. [SW92] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 4 (992), no. 4, 83 874. [Wil87] J. Michael Wilson, Weighted inequalities for the dyadic square function without dyadic A, Duke Math. J. 55 (987), no., 9 5. Department of Mathematics and Statistics, University of Helsinki, P.O.. 68, FI-4 Helsinki, Finland E-mail address: tuomas.hytonen@helsinki.fi Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 48 Sevilla, Spain E-mail address: carlosperez@us.es Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 48 Sevilla, Spain E-mail address: erela@us.es