A new quantitative two weight theorem for the Hardy-Littlewood maximal operator
aa r X i v : . [ m a t h . C A ] M a y A NEW QUANTITATIVE TWO WEIGHT THEOREM FORTHE HARDY-LITTLEWOOD MAXIMAL OPERATOR
CARLOS P´EREZ AND EZEQUIEL RELA
Abstract.
A quantitative two weight theorem for the Hardy-Little-wood maximal operator is proved improving the known ones. As aconsequence a new proof of the main results in [HP] and [HPR12] isobtained which avoids the use of the sharp quantitative reverse Holderinequality for A ∞ proved in those papers. Our results are valid withinthe context of spaces of homogeneous type without imposing the non-empty annuli condition. Introduction and Main results
Introduction.
The purpose of this note is to present a quantitative two weight theorem for the Hardy-Littlewood maximal operator when theunderlying space is a space of homogeneous type S (SHT in the sequel),endowed with a quasimetric ρ and a doubling measure µ (see Section 2.1 forthe precise definitions). We briefly recall some background on this problemin the euclidean or classical setting, when we are working in R n and weconsider Lebesgue measure and euclidean metric. We also assume that inthis classical setting all the maximal operators involved and A p classes ofweights are defined over cubes. Let M stand for the usual uncentered Hardy-Littlewood maximal operator: M f ( x ) = sup Q ∋ x | Q | Z Q | f | dx. The problem of characterizing the pair of weights for which the maximaloperator is bounded between weighted Lebesgue spaces was solved by Sawyer[Saw82]: To be more precise, if 1 < p < ∞ we define for any pair of weights w, σ , the (two weight) norm,(1.1) k M ( · σ ) k L p ( w ) := sup f ∈ L p ( σ ) k M ( f σ ) k L p ( w ) k f k L p ( σ ) Mathematics Subject Classification.
Primary: 42B25. Secondary: 43A85.
Key words and phrases.
Two weight theorem, Space of homogeneous type, Mucken-houpt weights, Calder´on-Zygmund, Maximal functions.Both authors are supported by the Spanish Ministry of Science and Innovation grantMTM2012-30748 and by the Junta de Andaluc´ıa, grant FQM-4745. then Sawyer showed that k M ( · σ ) k L p ( w ) is finite if and and only ifsup Q R Q ( M ( χ Q σ ) p wdxσ ( Q ) < ∞ , where the supremum is taken over all the cubes in R n . A quantitative preciseversion of this result is the following: if we define[ w, σ ] S p := (cid:18) σ ( Q ) Z Q M ( σχ Q ) p w dx (cid:19) /p . then(1.2) k M ( · σ ) k L p ( w ) ∼ p ′ [ w, σ ] S p , where p + p ′ = 1. This result is due to K. Moen and can be found in [Moe09].However, it is still an open problem to find a characterization more closelyrelated to the A p condition of Muckenhoupt which is easier to use in appli-cations. Indeed, recall that the two weight A p condition:sup Q (cid:18) − Z Q w dx (cid:19) (cid:18) − Z Q v − p − dx (cid:19) p − < ∞ is necessary for the boundedness of M from L p ( v ) into L p ( w ) (which isclearly equivalent, setting σ = v − p ′ , to the two weight problem), but it isnot sufficient. Therefore, the general idea is to strengthen the A p conditionto make it sufficient. The first result on this direction is due to Neugebauer[Neu83], proving that, for any r >
1, it is sufficient to consider the following“power bump” for the A p condition:(1.3) sup Q (cid:18) − Z Q w r dx (cid:19) r (cid:18) − Z Q v − rp − dx (cid:19) ( p − r < ∞ Later, the first author improved this result in [P´er95] by considering adifferent approach which allows to consider much larger classes weights. Thenew idea is to replace only the average norm associated to the weight v − p − in (1.3) by an “stronger” norm which is often called a “bump”. This normis defined in terms of an appropriate Banach function X space satisfyingcertain special property. This property is related to the L p boundedness ofa natural maximal function related to the space. More precisely, for a givenBanach function space X , the local X -average of a measurable function f associated to the cube Q is defined as k f k X,Q = (cid:13)(cid:13) τ ℓ ( Q ) ( f χ Q ) (cid:13)(cid:13) X , where τ δ is the dilation operator τ δ f ( x ) = f ( δx ), δ > ℓ ( Q ) stands forthe sidelength of the cube Q . The natural maximal operator associated tothe space X is defined as M X f ( x ) = sup Q : x ∈ Q k f k X,Q
UANTITATIVE TWO WEIGHT THEOREM 3 and the key property is that the maximal operator M X ′ is bounded on L p ( R n ) where X ′ is the associate space to X (see (1.5) below).As a corollary of our main result, Theorem 1.3, we will give a quantitativeversion of the main result from [P´er95] regarding sufficient conditions for thetwo weight inequality to hold: Theorem 1.1.
Let w and σ be a pair of weights that satisfies the condition (1.4) sup (cid:18) − Z Q w dx (cid:19) k σ /p ′ k pX,Q < ∞ . Suppose, in addition, that the maximal operator associated to the associatespace is bounded on L p ( R n ) : (1.5) M X ′ : L p ( R n ) → L p ( R n ) . Then there is a finite positive constant C such that: k M ( · σ ) k L p ( w ) ≤ C. In this note we give a different result of this type with the hope that itmay lead to different, possible better, conditions for the two weight problemfor Singular Integral Operators.Most of the interesting examples are obtained when X is an Orlicz space L Φ defined in term of the Young function Φ (see Section 2.1 for the precisedefinitions). In this case, the local average with respect to Φ over a cube Q is k f k Φ ,Q = k f k Φ ,Q,µ = inf (cid:26) λ > µ ( Q ) Z Q Φ (cid:18) | f | λ (cid:19) dx ≤ (cid:27) where µ is here the Lebesgue measure. The corresponding maximal functionis(1.6) M Φ f ( x ) = sup Q : x ∈ Q k f k Φ ,Q . Related to condition (1.4) we introduce here the following quantities.
Definition 1.2.
Let ( S , dµ ) be a SHT. Given a ball B ⊂ S , a Young functionΦ and two weights w and σ , we define the quantity(1.7) A p ( w, σ, B, Φ) := (cid:18) − Z B w dµ (cid:19) k σ /p ′ k p Φ ,B and we say that a pair of weights belong to the A p, Φ class if[ w, σ, Φ] A p := sup B A p ( w, σ, B, Φ) < ∞ , where the sup is taken over all balls in the space. In the particular case ofΦ( t ) = t p ′ , this condition corresponds to the classical A p condition and weuse the notation [ w, σ ] A p := sup B (cid:18) − Z B w dµ (cid:19) (cid:18) − Z B σ dµ (cid:19) p − . CARLOS P´EREZ AND EZEQUIEL RELA
We define now a generalization of the Fuji-Wilson constant of a A ∞ weight σ as introduced in [HP] by means of a Young function Φ:[ σ, Φ] W p := sup B σ ( B ) Z B M Φ (cid:16) σ /p χ B (cid:17) p dµ Note that the particular choice of Φ p ( t ) := t p reduces to the A ∞ constant( see Definition (2.5) from Section 2.1):(1.8) [ σ, Φ p ] W p = sup B σ ( B ) Z B M ( σχ B ) dµ = [ σ ] A ∞ . Main results.
Our main purpose in the present note is to address theproblem mentioned above within the context of spaces of homogeneous type.In this context, the Hardy–Littlewood maximal operator M is defined overballs:(1.9) M f ( x ) = sup B ∋ x µ ( B ) Z B | f | dµ. The Orlicz type maximal operators are defined also with balls and withrespect to the measure µ in the natural way.Our main result is the following theorem. Theorem 1.3.
Let < p < ∞ and let Φ be any Young function withconjugate function ¯Φ . Then, for any pair of weights w, σ , there exists astructural constant C > such that the (two weight) norm defined in (1.1) satisfies (1.10) k M ( · σ ) k L p ( w ) ≤ Cp ′ (cid:0) [ w, σ, Φ] A p [ σ, ¯Φ] W p (cid:1) /p , We emphasize that (1.10), which is even new in the usual context ofEuclidean Spaces, fits into the spirit of the A p − A ∞ theorem derived in[HP] and [HPR12]. The main point here is that we have a two weight resultwith a better condition and with a proof that avoids completely the use ofthe sharp quantitative reverse H¨older inequality for A ∞ weights proved inthese papers. This property is, of course, of independent interest but it isnot used in our results.From this Theorem, we derive several corollaries. First, we have a directproof of the two weight result derived in [HP] using the [ w ] A ∞ constant ofFujii-Wilson (2.5). Corollary 1.4.
Under the same hypothesis of Theorem 1.3, we have thatthere exists a structural constant
C > such that (1.11) k M ( · σ ) k L p ( w ) ≤ Cp ′ (cid:0) [ w, σ ] A p [ σ ] A ∞ (cid:1) /p . Note that the result in Theorem 1.3 involves two suprema like in Corollary1.4. It would interesting to find out if there is a version of this result involvingonly one supremum. There is some evidence that it could be the case, seefor example [HP], Theorem 4.3. See also the recent work [LM].As a second consequence of Theorem 1.3, we have the announced quanti-tative version of Theorem 1.1:
UANTITATIVE TWO WEIGHT THEOREM 5
Corollary 1.5.
Under the same hypothesis of Theorem 1.3, we have thatthere exists a structural constant
C > such that k M ( · σ ) k L p ( w ) ≤ Cp ′ [ w, σ, Φ] /pA p k M ¯Φ k L p ( R n ) We remark that this approach produces a non-optimal dependence on p ,since we have to pay with one p ′ for using Sawyer’s theorem. However, theideas from the proof of Theorem 1.3 can be used to derive a direct proof ofCorollary 1.5 without the p ′ factor. We include the proof in the appendix.Finally, for the one weight problem, we recover the known mixed bound. Corollary 1.6.
For any A p weight w the following mixed bound holds: k M k L p ( w ) ≤ Cp ′ (cid:0) [ w ] A p [ σ ] A ∞ (cid:1) /p where C is an structural constant and as usual σ = w − p ′ is the dual weight.Remark . To be able to extend the proofs to this general scenario, we needto use (and prove) suitable versions of classical tools on this subject, suchas Calder´on–Zygmund decompositions. We remark that in previous works([PW01], [SW92]) most of the results are proved under the assumption thatthe space has non-empty annuli. The main consequence of this property isthat in that case the measure µ enjoys a reverse doubling property, whichis crucial in the proof of Calder´on–Zygmund type lemmas. However, thisassumption implies, for instance, that the space has infinite measure andno atoms (i.e. points with positive measure) and therefore constraints thefamily of spaces under study. Recently, some of those results were provenwithout this hypothesis, see for example [PS04]. Here we choose to workwithout the annuli property and therefore we need to adapt the proofs from[PW01]. Hence, we will need to consider separately the cases when the spacehas finite or infinite measure. An important and useful result on this matteris the following: Lemma 1.8 ([GGKK98]) . Let ( S , ρ, µ ) be a space of homogeneous type.Then S is bounded if and only if µ ( S ) < ∞ . Outline.
The article is organized as follows. In Section 2 we summarizesome basic needed results on spaces of homogeneous type and Orlicz spaces.We also include a Calder´on–Zygmund type decomposition lemma. In Section3 we present the proofs of our results. Finally, we include in Section 4 anAppendix with a direct proof of a slightly better result than Corollary 1.4.2. preliminaries
In this section we first summarize some basic aspects regarding spacesof homogeneous type and Orlicz spaces. Then, we include a Calder´on–Zygmund (C–Z) decomposition lemma adapted to our purposes.
CARLOS P´EREZ AND EZEQUIEL RELA
Spaces of homogeneous type.
A quasimetric d on a set S is a func-tion d : S × S → [0 , ∞ ) which satisfies(1) d ( x, y ) = 0 if and only if x = y ;(2) d ( x, y ) = d ( y, x ) for all x, y ;(3) there exists a finite constant κ ≥ x, y, z ∈ S , d ( x, y ) ≤ κ ( d ( x, z ) + d ( z, y )) . Given x ∈ S and r >
0, we define the ball with center x and radius r , B ( x, r ) := { y ∈ S : d ( x, y ) < r } and we denote its radius r by r ( B ) and itscenter x by x B . A space of homogeneous type ( S , d, µ ) is a set S endowedwith a quasimetric d and a doubling nonnegative Borel measure µ such that(2.1) µ ( B ( x, r )) ≤ Cµ ( B ( x, r ))Let C µ be the smallest constant satisfying (2.1). Then D µ = log C µ iscalled the doubling order of µ . It follows that(2.2) µ ( B ) µ ( ˜ B ) ≤ C κµ (cid:18) r ( B ) r ( ˜ B ) (cid:19) D µ for all balls ˜ B ⊂ B. In particular for λ > B a ball, we have that(2.3) µ ( λB ) ≤ (2 λ ) D µ µ ( B ) . Here, as usual, λB stands for the dilation of a ball B ( x, λr ) with λ > c = c ( κ, µ ) > structural constant if it depends only on the quasimetric constant κ and thedoubling constant C µ .An elementary but important property of the quasimetric is the following.Suppose that we have two balls B = B ( x , r ) and B = B ( x , r ) with nonempty intersection. Then,(2.4) r ≤ r = ⇒ B ⊂ κ (2 κ + 1) B . This is usually known as the “engulfing” property and follows directly fromthe quasitriangular property of the quasimetric.In a general space of homogeneous type, the balls B ( x, r ) are not nec-essarily open, but by a theorem of Macias and Segovia [MS79], there is acontinuous quasimetric d ′ which is equivalent to d (i.e., there are positiveconstants c and c such that c d ′ ( x, y ) ≤ d ( x, y ) ≤ c d ′ ( x, y ) for all x, y ∈ S )for which every ball is open. We always assume that the quasimetric d iscontinuous and that balls are open.We will adopt the usual notation: if ν is a measure and E is a measurableset, ν ( E ) denotes the ν -measure of E . Also, if f is a measurable functionon ( S , d, µ ) and E is a measurable set, we will use the notation f ( E ) := R E f ( x ) dµ . We also will denote the µ -average of f over a ball B as f B = − R B f dµ . We recall that a weight w (any non negative measurable function) UANTITATIVE TWO WEIGHT THEOREM 7 satisfies the A p condition for 1 < p < ∞ if[ w ] A p := sup B (cid:18) − Z B w dµ (cid:19) (cid:18) − Z B w − p − dµ (cid:19) p − , where the supremum is taken over all the balls in S . The A ∞ class is definedin the natural way by A ∞ := S p> A p This class of weights can also be characterized by means of an appropriateconstant. In fact, there are various different definitions of this constant, allof them equivalent in the sense that they define the same class of weights.Perhaps the more classical and known definition is the following due toHruˇsˇcev [Hru84] (see also [GCRdF85]):[ w ] expA ∞ := sup B (cid:18) − Z B w dµ (cid:19) exp (cid:18) − Z B log w − dµ (cid:19) . However, in [HP] the authors use a “new” A ∞ constant (which was originallyintroduced implicitly by Fujii in [Fuj78] and later by Wilson in [Wil87]),which seems to be better suited. For any w ∈ A ∞ , we define(2.5) [ w ] A ∞ := [ w ] WA ∞ := sup B w ( B ) Z B M ( wχ B ) dµ, where M is the usual Hardy–Littlewood maximal operator. When the un-derlying space is R d , it is easy to see that [ w ] A ∞ ≤ c [ w ] expA ∞ for some structural c >
0. In fact, it is shown in [HP] that there are examples showing that[ w ] A ∞ is much smaller than [ w ] expA ∞ The same line of ideas yields the inequal-ity in this wider scenario. See the recent work of Beznosova and Reznikov[BR] for a comprehensive and thorough study of these different A ∞ con-stants. We also refer the reader to the forthcoming work of Duoandikoetxea,Martin-Reyes and Ombrosi [DMRO13] for a discussion regarding differentdefinitions of A ∞ classes.2.2. Orlicz spaces.
We recall here some basic definitions and facts aboutOrlicz spaces.A function Φ : [0 , ∞ ) → [0 , ∞ ) is called a Young function if it is continu-ous, convex, increasing and satisfies Φ(0) = 0 and Φ( t ) → ∞ as t → ∞ . ForOrlicz spaces, we are usually only concerned about the behaviour of Youngfunctions for t large. The space L Φ is a Banach function space with theLuxemburg norm k f k Φ = k f k Φ ,µ = inf (cid:26) λ > Z S Φ( | f | λ ) dµ ≤ (cid:27) . Each Young function Φ has an associated complementary Young function ¯Φsatisfying t ≤ Φ − ( t ) ¯Φ − ( t ) ≤ t for all t >
0. The function ¯Φ is called the conjugate of Φ, and the space L ¯Φ is called the conjugate space of L Φ . For example, if Φ( t ) = t p for 1 < p < ∞ ,then ¯Φ( t ) = t p ′ , p ′ = p/ ( p − L p ( µ ) is L p ′ ( µ ). CARLOS P´EREZ AND EZEQUIEL RELA
A very important property of Orlicz spaces is the generalized H¨older in-equality(2.6) Z S | f g | dµ ≤ k f k Φ k g k ¯Φ . Now we introduce local versions of Luxemburg norms. If Φ is a Youngfunction, let k f k Φ ,B = k f k Φ ,B,µ = inf (cid:26) λ > µ ( B ) Z B Φ (cid:18) | f | λ (cid:19) dµ ≤ (cid:27) . Furthermore, the local version of the generalized H¨older inequality (2.6) is(2.7) 1 µ ( B ) Z B f g dµ ≤ k f k Φ ,B k g k ¯Φ ,B . Recall the definition of the maximal type operators M Φ from (1.6):(2.8) M Φ f ( x ) = sup B : x ∈ B k f k Φ ,B . An important fact related to this sort of operator is that its boundednessis related to the so called B p condition. For any positive function Φ (notnecessarily a Young function), we have that k M Φ k pL p ( S ) ≤ c µ,κ α p (Φ) , where α p (Φ) is the following tail condition(2.9) α p (Φ) = Z ∞ Φ( t ) t p dtt < ∞ . It is worth noting that in the recent article [LL] the authors define the appro-priate analogue of the B p condition in order to characterize the boundednessof the strong Orlicz-type maximal function defined over rectangles both inthe linear and multilinear cases. Recent developments and improvementscan also be found in [MP], where the authors addressed the problem ofstudying the maximal operator between Banach function spaces.2.3.
Calder´on–Zygmund decomposition for spaces of homogeneoustype.
The following lemma is a classical result in the theory, regarding a decom-position of a generic level set of the Hardy–Littlewood maximal function M .Some variants can be found in [AM84] for M and in [Aim85] for the centeredmaximal function M c . In this latter case, the proof is straightforward. Weinclude here a detailed proof for the general case of M where some extrasubtleties are needed. Lemma 2.1 (Calder´on–Zygmund decomposition) . Let B be a fixed ball andlet f be a bounded nonnegative measurable function. Let M be the usual noncentered Hardy–Littlewood maximal function. Define the set Ω λ as (2.10) Ω λ = { x ∈ B : M f ( x ) > λ } , UANTITATIVE TWO WEIGHT THEOREM 9
Let λ > be such that λ ≥ − R B f dµ . If Ω λ is non-empty, then given η > ,there exists a countable family { B i } of pairwise disjoint balls such that, for θ = 4 κ + κ , i) ∪ i B i ⊂ Ω λ ⊂ ∪ i θB i , ii) For all i , λ < µ ( B i ) Z B i f dµ. iii) If B is any ball such that B i ⊂ B for some i and r ( B ) ≥ ηr ( B i ) , wehave that (2.11) 1 µ ( ηB ) Z ηB f dµ ≤ λ. Proof.
Define, for each x ∈ Ω λ , the following set: R λx = (cid:26) r > − Z B f dµ > λ, x ∈ B = B ( y, r ) (cid:27) , which is clearly non-empty. The key here is to prove that R λx is bounded.If the whole space is bounded, there is nothing to prove. In the case ofunbounded spaces, we argue as follows. Since the space is of infinite measure(recall Lemma 1.8), and clearly S = S r> B ( x, r ), we have that µ ( B ( x, r ))goes to + ∞ when r → ∞ for any x ∈ S . Therefore, for K = κ (2 κ + 1), wecan choose r such that the ball B = B ( x, r ) satisfies the inequality µ ( B ) ≥ K ) D µ k f k L λ Suppose now that sup R λx = + ∞ . Then we can choose a ball B = B ( y, r )for some y such that x ∈ B , − R B f dµ > λ and r > r . Now, by theengulfing property (2.4) we obtain that B ⊂ KB . The doubling condition(2.3) yields µ ( B ) ≤ µ ( KB ) ≤ (2 k ) D µ µ ( B )Then we obtain that 2 k f k L λ ≤ µ ( B ) < k f k L λ which is a contradiction. We conclude that, in any case, for any x ∈ Ω λ , wehave that sup R λx < ∞ .Now fix η >
1. If x ∈ Ω λ , there is a ball B x containing x , whose radius r ( B x ) satisfies sup R λx η < r ( B x ) ≤ sup R λx , and for which − R B x f dµ > λ . Thusthe ball B x satisfies ii) and iii). Also note that Ω λ = S x ∈ Ω λ B x . Pickinga Vitali type subcover of { B x } x ∈ Ω λ as in [SW92], Lemma 3.3, we obtain afamily of pairwise disjoint balls { B i } ⊂ { B x } x ∈ Ω λ satisfying i). Therefore { B i } satisfies i), ii) and iii). (cid:3) We will need another important lemma, in order to handle simultaneouslydecompositions of level sets at different scales.
Lemma 2.2.
Let B be a ball and let f be a bounded nonnegative measurablefunction. Let also a ≫ and, for each integer k such that a k > − R B f dµ , wedefine Ω k as (2.12) Ω k = n x ∈ B : M f ( x ) > a k o , Let { E ki } i,k be defined by E ki = B ki \ Ω k +1 , where the family of balls { B ki } i,k is obtained by applying Lemma 2.1 to each Ω k . Then, for θ = 4 κ + κ as inthe previous Lemma and η = κ (4 κ + 3) , the following inequality holds: (2.13) µ ( B ki ∩ Ω k +1 ) < (4 θη ) D µ a µ ( B ki ) . Consequently, for sufficiently large a , we can obtain that (2.14) µ ( B ki ) ≤ µ ( E ki ) . Proof.
To prove the claim, we apply Lemma 2.1 with η = κ (4 κ + 3). Then,by part i), we have that, for θ = 4 κ + κ Ω k +1 ⊂ [ m θB k +1 m and then(2.15) µ ( B ki ∩ Ω k +1 ) ≤ X m µ ( B ki ∩ θB k +1 m ) . Suppose now that B ki ∩ θB k +1 m = ∅ . We claim that r ( B k +1 m ) ≤ r ( B ki ).Suppose the contrary, namely r ( B k +1 m ) > r ( B ki ). Then, by property (2.4),we can see that B ki ⊂ κ (4 κ + 3) B k +1 m = ηB k +1 m . For B = ηB k +1 m , part iii)from Lemma 2.1 gives us that the average satisfies(2.16) 1 µ ( B ) Z B f dµ ≤ a k . Now, by the properties of the family { B k +1 m } m and the doubling conditionof µ , we have that, for a > (2 η ) D µ ,(2.17) 1 µ ( ηB k +1 m ) Z ηB k +1 m f dµ > a k +1 (2 η ) D µ > a k . This last inequality contradicts (2.16). Then, whenever B ki ∩ θB k +1 m = ∅ , wehave that r ( B k +1 m ) ≤ r ( B ki ) and from that it follows that B k +1 m ⊂ ηB ki . Thesum (2.15) now becomes µ ( B ki ∩ Ω k +1 ) ≤ X m : B k +1 m ⊂ ηB ki µ ( B kj ∩ θB k +1 m ) ≤ (2 θ ) D µ X m : B k +1 m ⊂ ηB ki µ ( B k +1 m ) ≤ (2 θ ) D µ a k +1 Z ηB ki f dµ UANTITATIVE TWO WEIGHT THEOREM 11 since the sets { B k +1 m } m are pairwise disjoint. Finally, by part iii) of Lemma2.1, we obtain µ ( B ki ∩ Ω k +1 ) ≤ (4 θη ) D µ a µ ( B ki ) , which is inequality (2.13). (cid:3) Proofs of the main results
We present here the proof or our main results. Our starting point is aversion of the sharp two weight inequality (1.2) valid for SHT from [Kai]:
Theorem 3.1 ([Kai]) . Let ( S , ρ, µ ) a SHT. Then the H–L maximal operator M defined by (1.9) satisfies the bound (3.1) k M ( f σ ) k L p ( w ) ≤ Cp ′ [ w, σ ] S p k f k L p ( σ ) , where [ w, σ ] S p is the Sawyer’s condition with respect to balls: (3.2) [ w, σ ] S p := sup B (cid:18) σ ( B ) Z B M ( σχ B ) p w dµ (cid:19) /p . We now present the proof of the main result.
Proof of Theorem 1.3.
By Theorem 3.1, we only need to prove that[ w, σ ] S p ≤ C [ w, σ, Φ] /pA p [ σ, ¯Φ] /pW p for some constant C , for any Young function Φ, for any 1 < p < ∞ . Let B be a fixed ball B and consider the sets Ω k from (2.12) for the function σχ B for any k ∈ Z . We remark here that in order to apply a C–Z decompositionof these sets, we need the level of the decomposition to be larger that theaverage over the ball. We proceed as follows. Take any a > k ∈ Z such that(3.3) a k − < − Z B σ dµ ≤ a k . Now, let A be the set of the small values of the maximal function: A = (cid:26) x ∈ B : M ( σχ B ) ≤ a − Z B σ dµ (cid:27) . For any x ∈ B \ A , we have that M ( σχ B )( x ) > a − Z B σ dµ > a k ≥ − Z B σ dµ. Therefore, Z B M ( σχ B ) p w dµ = Z A M ( σχ B ) p w dµ + Z B \ A M ( σχ B ) p w dµ ≤ a p w ( B ) (cid:18) − Z B σ dµ (cid:19) p + X k ≥ k Z Ω k \ Ω k +1 M ( σχ B ) p w dµ = I + II The first term I can be bounded easily. By the general H¨older inequality(2.7), we obtain I ≤ a p (cid:18) − Z B w dµ (cid:19) k σ /p ′ k p Φ ,B k σ /p k p ¯Φ ,B µ ( B ) ≤ w, σ, Φ] A p Z B M ¯Φ ( σ /p χ B ) p dµ Now, for the second term II , we first note that Z B \ A M ( σχ B ) p w dµ = X k ≥ k Z Ω k \ Ω k +1 M ( σχ B ) p w dµ ≤ a p X k ≥ k a kp w (Ω k )By the choice of k , we can apply Lemma 2.1 to perform a C–Z decom-position at all levels k ≥ k and obtain a family of balls { B ki } i,k with theproperties listed in that lemma. Then, Z B \ A M ( σχ B ) p w dµ ≤ a p X k,i − Z B ki σχ B dµ ! p w ( θB ki ) ≤ a p X k,i µ ( θB ki ) µ ( B ki ) − Z θB ki σ p σ p ′ χ B dµ ! p w ( θB ki )We now proceed as before, using the local generalized Holder inequality (2.7)and the doubling property (2.3) of the measure (twice). Then we obtain Z B \ A M ( σχ B ) p wdµ ≤ a p (2 θ ) ( p +1) D µ [ w, σ, Φ] A p X k,i (cid:13)(cid:13)(cid:13) σ p χ B (cid:13)(cid:13)(cid:13) p ¯Φ ,θB ki µ ( B ki )The key here is to use Lemma 2.2 to pass from the family { B ki } to thepairwise disjoint family { E ki } . Then, for a ≥ θη ) D µ , we can bound thelast sum as follows X k,i (cid:13)(cid:13)(cid:13) σ p χ B (cid:13)(cid:13)(cid:13) p ¯Φ ,θB ki µ ( B ki ) ≤ X k,i (cid:13)(cid:13)(cid:13) σ p χ B (cid:13)(cid:13)(cid:13) p ¯Φ ,θB ki µ ( E ki ) ≤ X k,i Z E ki M ¯Φ ( σ p χ B ) p dµ ≤ Z B M ¯Φ ( σ p χ B ) p dµ since the sets { E k,j } are pairwise disjoint. Collecting all previous estimatesand dividing by σ ( B ), we obtain the desired estimate[ w, σ ] pS p ≤ a p (2 θ ) ( p +1) D µ [ w, σ, Φ] A p [ σ, ¯Φ] W p , UANTITATIVE TWO WEIGHT THEOREM 13 and the proof of Theorem 1.3 is complete. (cid:3)
It remains to prove Corollary 1.4. To that end, we need to consider thespecial case of Φ( t ) = t p ′ . Proof of Corollary 1.4.
Considering then Φ( t ) = t p ′ , the quantity (1.7) is A p ( w, σ, B, Φ) = (cid:18) − Z B w dµ (cid:19) k σ /p ′ k p Φ ,B = (cid:18) − Z B w ( y ) dµ (cid:19) (cid:18) − Z B σ dµ (cid:19) p − . In addition, we have from (1.8) that [ σ, Φ p ′ ] W p = [ σ, Φ p ] W p = [ σ ] A ∞ andtherefore we obtain (1.11). (cid:3) For the proof of Corollary 1.5, we simply use the boundedness of M ¯Φ on L p ( µ ), [ σ, ¯Φ] W p := sup B σ ( B ) Z B M ¯Φ (cid:16) σ /p χ B (cid:17) p dµ ≤ k M ¯Φ k pL p . The proof of Corollary 1.6 is trivial.4.
Appendix
We include here a direct proof of version of Corollary 1.5 which is better interms of the dependence on p . Precisely, we have the following Proposition. Proposition 4.1.
Let < p < ∞ . For any pair of weights w, σ and anyYoung function Φ , there exists a structural constant C > such that k M ( f σ ) k L p ( w ) ≤ C [ w, σ, Φ] /pA p k M ¯Φ k L p k f k L p ( σ ) Proof of Proposition 4.1.
By density it is enough to prove the inequalityfor each nonnegative bounded function with compact support f . We firstconsider the case of unbounded S . In this case we have − R S f σ dµ = 0.Therefore, instead of the sets from sets from (2.12), we considerΩ k = n x ∈ S : M ( f σ )( x ) > a k o , for any a > k ∈ Z . Then, we can write Z S M ( f σ ) p w dµ = X k Z Ω k \ Ω k +1 M ( f σ ) p w dµ Then, following the same line of ideas as in the proof of Theorem 1.3, weobtain Z S M ( f σ ) p wdµ ≤ a p (2 θ ) ( p +1) D µ [ w, σ, Φ] A p X k,i (cid:13)(cid:13)(cid:13) f σ p (cid:13)(cid:13)(cid:13) p ¯Φ ,θB ki µ ( B ki ) By Lemma 2.2 we can replace the family { B ki } by the pairwise disjoint family { E ki } to obtain the desired estimate: Z S M ( f σ ) p w dµ ≤ a p (2 θ ) ( p +1) D µ [ w, σ, Φ] A p k M ¯Φ k pL p Z S f p σ dµ. In the bounded case, the whole space is a ball and we can write S = B ( x, R )for any x and some R >
0. The problem here is to deal with the small valuesof λ , since we cannot apply Lemma 2.2 for a k ≤ − R S f σ dµ . We then takeany a > k ∈ Z to verify (3.3): a k − < − Z S f σ dµ ≤ a k and argue as in the proof of Theorem 1.3. (cid:3) Now, from this last proposition, we can derive another proof of the mixedbound (1.11) from Corollary 1.4. The disadvantage of this approach withrespect to the previous one is that we need a deep property of A ∞ weights:the sharp Reverse H¨older Inequality. In the whole generality of SHT, weonly know a weak version of this result from the recent paper [HPR12]: Theorem 4.2 (Sharp weak Reverse H¨older Inequality, [HPR12]) . Let w ∈ A ∞ . Define the exponent r ( w ) = 1 + τ κµ [ w ] A ∞ , where τ κµ is an structuralconstant. Then, (cid:18) − Z B w r ( w ) dµ (cid:19) /r ( w ) ≤ κ ) D µ − Z κB w dµ, where B is any ball in S . The other ingredient for the alternative proof of Corollary 1.4 is the knownestimate for the operator norm for M . For any 1 < q < ∞ , we have that k M k qL q ∼ q ′ . Another proof of Corollary 1.4.
Consider the particular choice of Φ( t ) = t p ′ r for r >
1. Then quantity (1.7) is A p ( w, σ, B, Φ) = (cid:18) − Z B w ( y ) dµ (cid:19) (cid:18) − Z B σ r dµ (cid:19) p/rp ′ If we choose r from the sharp weak reverse H¨older property (Theorem 4.2),we obtain that A p ( w, σ, B, Φ) = (cid:18) − Z B w dµ (cid:19) (cid:18) κ ) D µ − Z κB σ dµ (cid:19) p − ≤ p − (4 κ ) pD µ (cid:18) − Z κB w dµ (cid:19) (cid:18) − Z κB σ dµ (cid:19) p − ≤ p − (4 κ ) pD µ [ w, σ ] A p UANTITATIVE TWO WEIGHT THEOREM 15
And therefore the proof of Proposition 4.1 gives k M ( f σ ) k L p ( w ) ≤ C [ w, σ ] /pA p k M ¯Φ k L p ( S ,dµ ) k f k L p ( σ ) . We conclude with the proof by computing k M ¯Φ k L p for Φ( t ) = t p ′ r . We use(2.9), and then we obtain that k M ¯Φ k pL p ≤ cr ′ p ′ . But, by the choice of r , itfollows that r ′ ∼ [ σ ] A ∞ and we obtain (1.11). (cid:3) References [Aim85] Hugo Aimar. Singular integrals and approximate identities on spaces of ho-mogeneous type.
Trans. Amer. Math. Soc. , 292(1):135–153, 1985.[AM84] Hugo Aimar and Roberto A. Mac´ıas. Weighted norm inequalities for theHardy-Littlewood maximal operator on spaces of homogeneous type.
Proc.Amer. Math. Soc. , 91(2):213–216, 1984.[BR] Oleksandra Beznosova and Alexander Reznikov. Equivalent definitions ofdyadic Muckenhoupt and reverse H¨older classes in terms of Carleson se-quences, weak classes, and comparability of dyadic L log L and A ∞ constants.Preprint, arXiv:1201.0520 (2012).[DMRO13] Javier Duoandikoetxea, Francisco Mart´ın-Reyes, and Sheldy Ombrosi. On the A ∞ conditions for general bases. 2013. Private communication.[Fuj78] Nobuhiko Fujii. Weighted bounded mean oscillation and singular integrals. Math. Japon. , 22(5):529–534, 1977/78.[GCRdF85] Jos´e Garc´ıa-Cuerva and Jos´e L. Rubio de Francia.
Weighted norm inequalitiesand related topics , volume 116 of
North-Holland Mathematics Studies . North-Holland Publishing Co., Amsterdam, 1985.[GGKK98] Ioseb Genebashvili, Amiran Gogatishvili, Vakhtang Kokilashvili, andMiroslav Krbec.
Weight theory for integral transforms on spaces of homo-geneous type , volume 92 of
Pitman Monographs and Surveys in Pure andApplied Mathematics . Longman, Harlow, 1998.[HP] Tuomas Hyt¨onen and Carlos P´erez. Sharp weighted bounds involving A ∞ . Anal. PDE . (to appear).[HPR12] Tuomas Hyt¨onen, Carlos P´erez, and Ezequiel Rela. Sharp Reverse H¨olderproperty for A ∞ weights on spaces of homogeneous type. J. Funct. Anal. ,263(12):3883–3899, 2012.[Hru84] Sergei V. Hruˇsˇcev. A description of weights satisfying the A ∞ condition ofMuckenhoupt. Proc. Amer. Math. Soc. , 90(2):253–257, 1984.[Kai] Anna Kairema. Two-weight norm inequalities for potential type and maximaloperators in a metric space. Publicacions Mathemtiques. to appear (2012).[LL] Liguang Liu and Teresa Luque. A B p condition for the strong maximal func-tion. Trans. Amer. Math. Soc. (to appear).[LM] Andrei K. Lerner and Kabe Moen. Mixed A p - A ∞ estimates with one supre-mum. Preprint, arXiv:1212.0571 (2013).[Moe09] Kabe Moen. Sharp one-weight and two-weight bounds for maximal operators. Studia Math. , 194(2):163–180, 2009.[MP] Mieczys law Masty lo and Carlos P´erez. The maximal operators between ba-nach function spaces.
Indiana Univ. Math. J. (To appear).[MS79] Roberto A. Mac´ıas and Carlos Segovia. Lipschitz functions on spaces of ho-mogeneous type.
Adv. in Math. , 33(3):257–270, 1979.[Neu83] C. J. Neugebauer. Inserting A p -weights. Proc. Amer. Math. Soc. , 87(4):644–648, 1983. [P´er95] Carlos P´erez. On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted L p -spaces with differentweights. Proc. London Math. Soc. (3) , 71(1):135–157, 1995.[PS04] Gladis Pradolini and Oscar Salinas. Maximal operators on spaces of homoge-neous type.
Proc. Amer. Math. Soc. , 132(2):435–441 (electronic), 2004.[PW01] Carlos P´erez and Richard L. Wheeden. Uncertainty principle estimates forvector fields.
J. Funct. Anal. , 181(1):146–188, 2001.[Saw82] Eric T. Sawyer. A characterization of a two-weight norm inequality for max-imal operators.
Studia Math. , 75(1):1–11, 1982.[SW92] E. Sawyer and R. L. Wheeden. Weighted inequalities for fractional integralson Euclidean and homogeneous spaces.
Amer. J. Math. , 114(4):813–874, 1992.[Wil87] J. Michael Wilson. Weighted inequalities for the dyadic square function with-out dyadic A ∞ . Duke Math. J. , 55(1):19–50, 1987.
Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Univer-sidad de Sevilla, 41080 Sevilla, Spain
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