Proof of an extension of E. Sawyer's conjecture about weighted mixed weak-type estimates
aa r X i v : . [ m a t h . C A ] M a r PROOF OF AN EXTENSION OF E. SAWYER’S CONJECTUREABOUT WEIGHTED MIXED WEAK-TYPE ESTIMATES
KANGWEI LI, SHELDY OMBROSI, AND CARLOS P´EREZ
Abstract.
We show that if v ∈ A ∞ and u ∈ A , then there is a constant c dependingon the A constant of u and the A ∞ constant of v such that (cid:13)(cid:13)(cid:13) T ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c k f k L ( uv ) , where T can be the Hardy-Littlewood maximal function or any Calder´on-Zygmund op-erator. This result was conjectured in [IMRN, (30)2005, 1849–1871] and constitutesthe most singular case of some extensions of several problems proposed by E. Sawyerand Muckenhoupt and Wheeden. We also improve and extends several quantitativeestimates. Introduction and main results
The purpose of this paper is to prove some extensions of several conjectures formulatedby E. Sawyer in [Sa] where it is proved the following weighted weak type inequality forthe Hardy-Littlewood maximal function on the real line: if u, v ∈ A , then(1.1) (cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c u,v k f k L ( uv ) . This estimate is a highly non-trivial extension of the classical weak type (1 ,
1) inequalityfor the maximal operator due to the presence of the weight function v inside the distribu-tion set. These type of estimates are also referred as mixed weak-type estimates. Observethat if v = 1 this result is the well known estimate due to C. Fefferman-E. Stein theorem[FS1] which holds if and only if u ∈ A . Also, if u = 1, then the result also holds when v ∈ A by a simple argument. However, in this case the A condition is not necessaryand many other examples can be found (see below).In the general situation this estimate becomes more difficult. There are basically twomain obstacles. The first problem is that the product uv may be very singular. Forinstance, let u ( x ) = v ( x ) = | x | − / on the real line. Then, u and v are A weights but theproduct uv is not locally integrable. The second drawback is that the operator f → M ( fv ) v ,which can be seen as a perturbation of M by the weight v , changes dramatically the levelsets of M . For instance, it is not clear how to apply directly any covering lemma to { M ( fv ) v > t } , specially in the case v ∈ A ∞ . Mathematics Subject Classification.
Primary: 42B25, Secondary: 42B20.
Key words and phrases.
Sawyer’s conjecture, weak type inequalities, A and A ∞ weights.K.L. and C.P. are supported by the Basque Government through the BERC 2014-2017 program andby Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence ac-creditation SEV-2013-0323. , S. O. is supported by CONICET PIP 11220130100329CO, Argentina.C.P. is supported by Spanish Ministry of Economy and Competitiveness MINECO through the projectMTM2014-53850-P. The main motivation in [Sa] to prove (1.1) is due to the fact that it yields a new proofof the classical A p theorem of Muckenhoupt [M] assuming the factorization theorem for A p weights (see [J]). However, there are many situations where these mixed weak typeinequalities appear naturally (see below). In the same paper, E. Sawyer conjectured that(1.1) should hold for the Hilbert transform and for M but in higher dimensions. A positiveanswer to both conjectures was found in [CMP2]. In fact these conjectures were furtherextended in several directions, first to the case of Calder´on-Zygmund operators (even tothe case of the maximal singular integral) and second to a larger class of weights solving atonce some other interesting conjectures formulated earlier by Muckenhoupt and Wheedenin [MW].The method of proofs in [CMP2] is based on the combination of the following facts orresults:1) An extrapolation type result for A ∞ weights in the spirit of the results obtained in[CMP1] and [CGMP].2) The use of the Rubio de Francia method [CUMP] within the context of Lorentz spaces.3) The use of the R. Coifman-C. Fefferman estimate (see (1.8) below) relating singularintegrals and the maximal function in the L p ( w ) spaces with p ∈ (0 , A ∞ , not just A p (see [CoF]).4) Reduction the problem from singular integral operators to the dyadic maximal function.Further extensions of the conjectures formulated by E. Sawyer (and also by Muckenhoupt-Wheeden [MW]) were raised in [CMP2]. The most difficult one of these conjectures isto prove (1.1) assuming that v ∈ A ∞ since it is the largest class. Although some im-provements were made later in [OP], and some more precise quantitative estimates wereobtained in [OPR], the full conjecture remained open. The main purpose of this paper isto prove this conjecture and to derive some consequences.Our main result is the following. Theorem 1.2.
Let M be the Hardy-Littlewood maximal operator on R n and let u ∈ A and v ∈ A ∞ . Then there is a finite constant c depending on the A constant of u and the A ∞ constant of v such that (1.3) (cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c k f k L ( uv ) . We point out some observations. • We will see that it is enough to consider the dyadic maximal function instead of M . • The conditions u ∈ A and v ∈ A ∞ are weaker than u ∈ A and uv ∈ A ∞ (which isone of the assumptions in [CMP2]). Indeed, when u ∈ A , uv ∈ A ∞ is equivalent with v ∈ A ∞ ( u ) (see [CMP2, Remark 2.2]) and the latter implies v ( E ) v ( Q ) ≤ C (cid:16) u ( E ) u ( Q ) (cid:17) ε ≤ C (cid:16) | E || Q | (cid:17) εδ , E ⊂ Q. • If we let v = | x | nr , r >
1, then the inequality above is true even for ( u, M u ), namely(1.4) (cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c k f k L ( vMu ) u ≥ u . • Surprisingly, (1.4) is false in general if v ∈ A ∞ as can be found in [OP, Example 3.1].Further, it is false if M u is replaced by M u . Again, no assumption is made on u . ROOF OF AN EXTENSION OF SAWYER’S CONJECTURE 3 • However, (1.4) is false when r = 1 even in the case u = 1, see [AM]. • Even further, if u = 1 and v = M µ , where µ is a non-negative function or measure,(1.3) is false as can be found in [OPR, Section 5]. Recall that v = M δ ≈ | x | n , where δ is the Dirac measure. This shows that for the class of weights v such that v ǫ ∈ A ∞ for some small ǫ , (1.3) is false in general. However, it was proved in [OP] that the keyextrapolation theorem, similar to Lemma 1.9 below, holds for these class of weights. • If uv = 1 then the result is true whenever v ∈ A , but it is false in general when v ∈ A p \ A , p >
1, see [PR].In view of these positive and negative examples we establish the following conjecture.
Conjecture 1.5.
Let v be a weight such that (cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( v ) ≤ c v k f k L ( v ) . Then, there is a finite constant c depending on the A constant of u and c v above suchthat (cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c k f k L ( uv ) u ∈ A . We already mentioned earlier that the main motivation to study these type of estimatesin [Sa] is to produce a new proof of the boundedness of the maximal operator on L p ( w ), w ∈ A p , assuming the factorization theorem of A p weights. However, there are severalother interesting applications: · Multilinear estimates.
It was shown in [LOPTT] that the multilinear operator definedby Q mj =1 M f j is too big to be considered as the maximal operator controlling the mul-tilinear Calder´on-Zygmund operators. Instead, the pointwise smaller maximal operator M introduced in that paper is the correct one (we remit the reader to the paper for thedefinition). Nevertheless, this operator is interesting on its own and it was shown therethat it does satisfy sharp weighted weak-type estimates whose proof is based on the mixedweak-type inequalities derived in [CMP2]. To be more precise, if { w i } mi =1 is a family ofweights such that w i ∈ A , for all i = 1 , , ..., m , and if ν = (cid:16) Q mj =1 w j (cid:17) /m , then(1.6) k m Y j =1 M f j k L m , ∞ ( ν ) ≤ C m Y j =1 k f i k L ( w i ) . which is an extension of the classical result mentioned above due to C. Fefferman-E. Stein, k M f k L , ∞ ( u ) ≤ c k f k L ( u ) if and only if u ∈ A . The strong version of this result, namely k m Y j =1 M f j k L p ( ν ~w ) ≤ c m Y j =1 k f j k L pj ( Mw j ) , where ν ~w = Q mj =1 w p/p j j with p = p + · · · + p m , p i ∈ (1 , ∞ ), i = 1 , · · · , m , can beobtained directly from the multilinear H¨older’s inequality and the classical Fefferman-Stein’s inequality. However, we cannot repeat the same argument for the weak normresult (1.6) and therefore a proof is required. Indeed, this result is more difficult since we KANGWEI LI, SHELDY OMBROSI, AND CARLOS P´EREZ have to control the distribution set: (in the case m = 2) ν (cid:26) x ∈ R n : M f ( x ) M f ( x ) > (cid:27) = ν (cid:26) x ∈ R n : M f ( x ) > M f ( x ) (cid:27) . The key observation is that (
M f ) − ∈ A ∞ with constant independent of f and hence weare dealing with an estimate that fits within our context. · Commutators with BMO functions. L p estimates of the commutators of Coifman-Rochberg-Weiss [ b, T ] can be derived in a very effective way by means of the conjugationmethod considered in [CRW]. This method works when T is a bounded linear operatoron L ( w ), w ∈ A and b ∈ BM O . However, this method breaks down when consideringendpoint estimates. Indeed, the conjugation method is intimately related to the familyoperators { T w } w ∈ A p , p >
1, of the form f → T w ( f ) := w T (cid:18) fw (cid:19) These families of operators have the same structure as the ones we are considering in thepresent paper. We could consider the question of whether they are of weak type (1 , ,
1) in general with respect to the Lebesguemeasure. This behavior is the same as in the case of commutators, as was shown in [P].On the other hand, these operators T w , when T is a Calder´on-Zygmund operator, are ofweak type (1 ,
1) when w ∈ A . However, the A class of weights is excluded from themethod of conjugation.There is also another interesting connection between weighted mixed weak type in-equalities and Ergodic Theory. We remit the reader to Section 4 in [Ma] for details.As a corollary of the main theorem 1.2 we obtain the following result. Corollary 1.7.
Let M be the Hardy-Littlewood maximal operator and let T be any oper-ator such that for some p ∈ (0 , ∞ ) and for any w ∈ A ∞ , there is a constant c dependingon the A ∞ constant of w such that, (1.8) Z R n | T f ( x ) | p w ( x ) dx ≤ c Z R n M f ( x ) p w ( x ) dx, for any function f such that the left hand side is finite. Then the analogue of (1.3) holdsfor T , namely if v ∈ A ∞ and u ∈ A , there is a constant c depending on the A constantof u and the A ∞ constant v such that, (cid:13)(cid:13)(cid:13) T ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c k f k L ( uv ) . The proof of the corollary is a consequence of the following extrapolation type resultfor A ∞ weights that can be found in [CMP2, Theorem 1.7]. Lemma 1.9.
Let F be a family of ordered pairs of non-negative, measurable functions ( f, g ) . Let p ∈ (0 , ∞ ) such that for every w ∈ A ∞ , (1.10) k f k L p ( w ) ≤ c k g k L p ( w ) ROOF OF AN EXTENSION OF SAWYER’S CONJECTURE 5 for all ( f, g ) ∈ F such that the left-hand side is finite, and where c depends only on the A ∞ constant of w . Then for all weights u ∈ A and v ∈ A ∞ , (cid:13)(cid:13)(cid:13) fv (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c (cid:13)(cid:13)(cid:13) gv (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) , ( f, g ) ∈ F , with a constant c depending on the A constant of u and the A ∞ constant of v . This result was extended in [OP] to a larger class of weights v , namely those weightssuch that for some δ > v δ ∈ A ∞ . However, as we already mentioned, it is not true thatfor every weight in this class, Theorem 1.2 holds.Corollary 1.7 also applies to the following operators:1) To any Calder´on-Zygmund operators. This follows from the classical good- λ inequalitybetween T and M due to R. Coifman-C. Fefferman [CoF].2) To any rough singular integral T Ω with Ω ∈ L ∞ ( S n − ) .
3) To the Bochner-Riesz multiplier at the critical index.There is no available good- λ inequality between any of these operators and M . However,an estimate like (1.8) holds for these operators as we have proved recently in [LPRR] forany p > g ∗ λ ( f ) (see [St] for the definition). In fact, (1.8) followsfrom M δ ( g ∗ λ ( f ))( x ) ≤ C δ,λ M f ( x ) λ > , < δ < , which can be found in [CP], together with the C. Fefferman-Stein estimate [FS2] (see also[Duo]):(1.11) Z R n | f ( x ) | p w ( x ) dx ≤ c Z R n M f ( x ) p w ( x ) dx, for any A ∞ weight w , any p , 0 < p < ∞ and for any function f such that left hand sideis finite.Another consequence of Theorem 1.2 is the following vector-valued extension. Corollary 1.12.
Let T be any operator satisfying the hypotheses from Corollary 1.7 aboveand let u ∈ A and v ∈ A ∞ . Let q ∈ (1 , ∞ ) then the following vector–valued extensionholds: there is a constant c depending on the A constant of u and the A ∞ constant v , (cid:13)(cid:13)(cid:13)(cid:13) (cid:16) P j | T ( vf j ) | q (cid:17) q v (cid:13)(cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c (cid:13)(cid:13)(cid:16) X j | f j | q (cid:17) q (cid:13)(cid:13) L ( uv ) . Proof of Theorem 1.2
This section is devoted to prove Theorem 1.2. We still follow the general idea of Sawyer[Sa]. However, Sawyer’s proof depends heavily on v ∈ A . To overcome this difficulty wecombine the ‘pigeon-hole’ technique, the Calder´on-Zygmund decomposition, and the twokey Lemmas 2.2 and 2.4 below.We start by using the well known fact that M ( f ) ≤ c n n X i =1 M D ( i ) f, KANGWEI LI, SHELDY OMBROSI, AND CARLOS P´EREZ where D ( i ) is a dyadic system for all 1 ≤ i ≤ n . So it is enough to obtain the followinginequality for any g bounded with compact support(2.1) uv { x ∈ R n : 1 < M d ( g )( x ) v ( x ) ≤ } ≤ C u,v Z R n gudx, where M d := M D and D is one of the dyadic system D ( i ) , 1 ≤ i ≤ n . We decompose theleft hand side of (2.1) as X k ∈ Z uv { x ∈ R n : 1 < M d ( g )( x ) v ( x ) ≤ , a k < v ( x ) ≤ a k +1 } := X k ∈ Z uv ( E k ) , where a > n . For each k we defineΩ k := { x ∈ R n : M d ( g )( x ) > a k } . Observe that E k ⊂ Ω k . Let Q k := { I kj } denote the collection of maximal, disjoint dyadiccubes whose union is Ω k . By maximality, a k < h g i I kj ≤ n a k . We split now the family of cubes { I kj } j as Q l,k := { I kj ∈ Q k : a k + l ≤ h v i I kj < a k + l +1 } , l ≥ Q − ,k := { I kj ∈ Q k : h v i I kj < a k } . Now, for a fixed I kj ∈ Q − ,k , since a k > h v i I kj , we form the Calder´on-Zygmund decom-position to vχ I kj at height a k . Hence, we obtain a collection of subcubes { I kj,i } i ∈ D ( I kj )which satisfy a k < h v i I kj,i < n a k , ∀ i. Moreover, v ( x ) ≤ a k , x ∈ I kj \ [ i I kj,i . Now denote Ω − ,k = S I kj ∈ ˜Ω − ,k S i I kj,i . We have X k ∈ Z uv ( E k ) = X k ∈ Z uv ( E k ∩ Ω k ) = X k ∈ Z X j uv ( E k ∩ I kj ) = ≤ X k ∈ Z X l ≥ X I kj ∈Q l,k a k +1 u ( E k ∩ I kj ) + X k ∈ Z X I kj ∈Q − ,k a k +1 u ( E k ∩ I kj ) ≤ X k ∈ Z X l ≥ X I kj ∈ Γ l,k a k +1 u ( E k ∩ I kj ) + X k ∈ Z X i : I kj,i ∈ Γ − ,k a k +1 u ( I kj,i ) , where Γ l,k = { I kj ∈ Q l,k : | I kj ∩ { x : a k < v ≤ a k +1 }| > } if l ≥ , and Γ − ,k = { I kj,i ∈ Q − ,k : | I kj,i ∩ { x : a k < v ≤ a k +1 }| > } . ROOF OF AN EXTENSION OF SAWYER’S CONJECTURE 7
In the following, we shall deal with the case l = − l ≥ X k ≥ N X l ≥ X I kj ∈ Γ l,k a k +1 u ( E k ∩ I kj ) + X k ≥ N X i : I kj,i ∈ Γ − ,k a k +1 u ( I kj,i ) , where N <
0. The following lemma is a key.
Lemma 2.2.
Γ = ∪ l ≥− ∪ k ≥ N Γ l,k is sparse.Proof. First, we prove that if I kj ( I ts or I kj,i ( I ts or I kj ( I ts,ℓ or I kj,i ( I ts,ℓ , then k > t . Thecase I kj ( I ts is obvious since they are maximal dyadic cubes in Q k and Q t , respectively.For the case I kj ( I ts,ℓ , then again I kj ( I ts . For the case I kj,i ( I ts , obviously I kj = I ts . Noticethat if I ts ( I kj , then t > k and h v i I ts > a t > a k which means there is some cube Q such that I kj,i ( Q and a k < h v i Q < n a k , a contradic-tion! Finally, if I kj,i ( I ts,ℓ , assume t > k , then I ts ( I kj . Therefore h v i I ts,ℓ > a t > a k , and I ts,ℓ ( I kj , again, a contradiction. So we have proved the first claim. With this claimat hand, it is easy to check that | [ Q ′ ,Q ∈ Γ Q ′ ( Q Q ′ | ≤ n a | Q | . (cid:3) The case l ≥ . First we need the following lemma.
Lemma 2.3.
Let w ∈ A ∞ , and let r w = 1 + τ n [ w ] A ∞ . Then for any cube Q (cid:18) | Q | Z Q w r w (cid:19) /r w ≤ | Q | Z Q w. As a consequence we have that for any cube Q and for any measurable set E ⊂ Qw ( E ) w ( Q ) ≤ (cid:18) | E || I | (cid:19) ǫ w , where ǫ w = τ n [ w ] A ∞ . The proof of this reverse H¨older inequality can be found in [HyPe1] and the consequenceis an application of H¨older’s inequality.Another key point is the following lemma.
Lemma 2.4.
For l ≥ and I kj ∈ Γ l,k , there exist constants c and c depends on u, v such that u ( E k ∩ I kj ) ≤ c e − c l u ( I kj ) . KANGWEI LI, SHELDY OMBROSI, AND CARLOS P´EREZ
Proof.
Since v ∈ A ∞ , by embedding, we know that there exists some q such that v ∈ A q .Then (cid:16) | E k ∩ I kj || I kj | (cid:17) q − a − k − ≤ (cid:16) | I kj | Z I kj v − q − (cid:17) q − ≤ [ v ] A q h v i I kj ≤ a − k − l [ v ] A q . It follows that | E k ∩ I kj || I kj | ≤ a − lq − [ v ] q − A q . Then since u ∈ A we can use Lemma 2.3 to get, u ( E k ∩ I kj ) u ( I kj ) ≤ c e − c l . (cid:3) Now return to the proof. Fix l , form the principal cubes for ∪ k ≥ N Γ l,k : let P l be themaximal cubes in ∪ k ≥ N Γ l,k , then for m ≥
0, if I ts ∈ P lm , we say I kj ∈ P lm +1 if I kj is maximal(in the sense of inclusion) in D ( I ts ) such that h u i I kj > h u i I ts Denote P l = ∪ m ≥ P lm and π ( Q ) is the minimal principal cube which contains Q . We have X k X l ≥ X I kj ∈ Γ l,k a k +1 u ( E k ∩ I kj ) Lemma . ≤ X l ≥ c e − c l a − l X k X I kj ∈ Γ l,k h v i I kj u ( I kj )= X l ≥ c e − c l a − l X k X I kj ∈ Γ l,k h u i I kj v ( I kj ) ≤ X l ≥ c e − c l a − l X I ts ∈P l h u i I ts X k,j : π ( I kj )= I ts v ( I kj ) Lemma . . n X l ≥ c e − c l a − l [ v ] A ∞ X I ts ∈P l h u i I ts v ( I ts ) ≤ a X l ≥ c e − c l [ v ] A ∞ X I ts ∈P l a t u ( I ts ) ≤ a X l ≥ c e − c l [ v ] A ∞ X I ts ∈P l h g i I ts u ( I ts ) ≤ a X l ≥ c e − c l [ v ] A ∞ Z R n g ( x ) X I ts ∈P l h u i I ts χ I ts ( x ) dx. For fixed x , there is a chain of principal cubes which contain x , say I mx . By the definitionof principal cubes, h u i I mx forms a geometric sequence (indeed, this sequence is finite), wehave X I ts ∈P l h u i I ts χ I ts ( x ) = X ≤ m ≤ m h u i I mx ≤ X ≤ m ≤ m m − m h u i I m x ≤ u ] A u ( x ) . ROOF OF AN EXTENSION OF SAWYER’S CONJECTURE 9
Finally, take the sum over l we obtain X k ∈ Z X l ≥ X I kj ∈ Γ l,k a k +1 u ( E k ∩ I kj ) ≤ c n,u,v [ u ] A [ v ] A ∞ k g k L ( u ) . The case l = − . In general, this case follows the strategy in [Sa]. But instead ofusing A property of v , the sparsity of Γ plays an important role. First of all, we definethe principal cubes with respect to u . Set F be the maximal cubes in Γ − ,N . Now for m ≥
0, assume that I ts,ℓ ∈ F m , then we say I kj,i ∈ F m +1 if I kj,i is maximal in D ( I ts,ℓ ) suchthat h u i I kj,i > a ( k − t ) δ h u i I ts,ℓ . Finally, we define F = ∪ m ≥ F m . We still denote by π ( Q ) the minimal principal cubewhich contains Q . If π ( I k ′ j ′ ,i ′ ) = I ts,ℓ , then by the proof of Lemma 2.2, k ′ ≥ t . And bydefinition, h u i I k ′ j ′ ,i ′ ≤ a ( k ′ − t ) δ h u i I ts,ℓ . We have X k ∈ Z X I kj,i ∈ Γ − ,k a k +1 u ( I kj,i ) ≤ a X k ∈ Z X I kj,i ∈ Γ − ,k h v i I kj,i u ( I kj,i ) ≤ a X I ts,ℓ ∈F h u i I ts,ℓ X k,j,i : π ( I kj,i )= I ts,ℓ a ( k − t ) δ v ( I kj,i )= a X I ts,ℓ ∈F h u i I ts,ℓ X k ≥ t a ( k − t ) δ X j,i : π ( I kj,i )= I ts,ℓ v ( I kj,i ) . By the sparsity, X j,i : π ( I kj,i )= I ts,ℓ | I kj,i | ≤ ( 2 n a ) ( k − t ) | I ts,ℓ | . It follows from Lemma 2.3 that X j,i : π ( I kj,i )= I ts,ℓ v ( I kj,i ) ≤ ( 2 n a ) k − t τn [ v ] A ∞ v ( I ts,ℓ ) . Let δ = 1 / ( c ′ n [ v ] A ∞ ), where c ′ n is a sufficient large constant which depends only on dimen-sion. Then we have X k ≥ t a ( k − t ) δ X j,i : π ( I kj,i )= I ts,ℓ v ( I kj,i ) ≤ c n [ v ] A ∞ v ( I ts,ℓ ) . It remains to estimate X I ts,ℓ ∈F h u i I ts,ℓ v ( I ts,ℓ ) ≤ X I ts,ℓ ∈F a t +1 u ( I ts,ℓ ) ≤ a X I ts,ℓ ∈F h g i I ts u ( I ts,ℓ )= a Z R n g ( x ) X I ts,ℓ ∈F | I ts | − u ( I ts,ℓ ) χ I ts ( x ) dx. We need to prove that(2.5) X I ts,ℓ ∈F | I ts | − u ( I ts,ℓ ) χ I ts ( x ) ≤ c n [ u ] A [ v ] A ∞ u ( x ) . To this end, let’s fix x . Since for fixed t , there is at most one I ts (keep in mind h v i I ts ≤ a t )contains x , if such I ts exists, we denote it by I t . Denote G = { I t : I t ∋ x } and form the principal cubes for G : set G = { I k } to be the maximal cube in G ( I k exists since k ≥ N ). Then if I k m ∈ G m , we say I k m +1 ∈ G m +1 if I k m +1 ⊂ I k m is maximalsuch that h u i I km +1 > h u i I km . Now we have X I ts,ℓ ∈F | I ts | − u ( I ts,ℓ ) χ I ts ( x ) = X I ts,ℓ ∈F | I t | − u ( I ts,ℓ )= X I ts,ℓ ∈F h u i I t u ( I ts,ℓ ) u ( I t ) ≤ X m h u i I km X I t ∈ Gk m ≤ t
Thus, X s,ℓ : I ts,ℓ ∈F u ( I ts,ℓ ) ≤ u ( { y ∈ I t : u ( y ) > λ } ) Lemma . ≤ u ( I t ) (cid:18) |{ y ∈ I t : u ( y ) > λ }|| I t | (cid:19) τn [ u ] A ≤ u ( I t ) (cid:0) λ − h u i I t (cid:1) cn [ u ] A . n u ( I t ) a t − kmcn [ u ] A v ] A ∞ . Finally, take the sum over t we conclude the proof of (2.6).3. Proof of Corollary 1.12
The proof of Corollary 1.12 follows from the following result which can be found in[CMP1].
Lemma 3.1.
Let F be a family of ordered pairs of non-negative, measurable functions ( f, g ) . Let p ∈ (0 , ∞ ) such that for every w ∈ A ∞ , Z R n f ( x ) p w ( x ) dx ≤ c Z R n g ( x ) p w ( x ) dx, ( f, g ) ∈ F , for all ( f, g ) ∈ F such that the left-hand side is finite, and where c depends only on the A ∞ constant of w . Then for all p, q ∈ (0 , ∞ ) , and w ∈ A ∞ , there is a constant c dependingon the A ∞ constant of w such that, (cid:13)(cid:13)(cid:13)(cid:16) X j ( f j ) q (cid:17) q (cid:13)(cid:13)(cid:13) L p ( w ) ≤ c (cid:13)(cid:13)(cid:13)(cid:16) X j ( g j ) q (cid:17) q (cid:13)(cid:13)(cid:13) L p ( w ) , { ( f j , g j ) } j ⊂ F . Now, the hypothesis (1.8) is satisfied for some p , namely Z R n | T f ( x ) | p w ( x ) dx ≤ c Z R n M f ( x ) p w ( x ) dx, and hence by Lemma 3.1, for all 0 < p, q < ∞ , and w ∈ A ∞ , (cid:13)(cid:13)(cid:13)(cid:16) X j | T ( f j ) | q (cid:17) q (cid:13)(cid:13)(cid:13) L p ( w ) ≤ C (cid:13)(cid:13)(cid:13)(cid:16) X j ( M f j ) q (cid:17) q (cid:13)(cid:13)(cid:13) L p ( w ) , for any for any vector function f = { f j } j such that the left hand side is finite. Now inthe case q > < q < ∞ and 0 < δ <
1, then there exists a constant c > q, δ, n such that for any vector function f = { f j } j M δ (cid:16) M q f (cid:17) ( x ) ≤ c M ( k f k ℓ q )( x ) x ∈ R n , using the notation M q f ( x ) = ( P ∞ i =1 M f j ( x ) q ) /q . Hence, for q > p ∈ (0 , ∞ ), (cid:13)(cid:13)(cid:13)(cid:16) X j | T ( f j ) | q (cid:17) q (cid:13)(cid:13)(cid:13) L p ( w ) ≤ c (cid:13)(cid:13)(cid:13) M ( k f k ℓ q ) (cid:13)(cid:13)(cid:13) L p ( w ) , by using (1.11). We can apply now Lemma 1.9. Indeed, hypthesis (1.10) is satisfiedchoosing f as (cid:16) P j | T ( f j ) | q (cid:17) q and g as M ( k f k ℓ q ). Hence, if u ∈ A and v ∈ A ∞ , (cid:13)(cid:13)(cid:13) (cid:16) P j | T ( f j ) | q (cid:17) q v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c k M ( k f k ℓ q ) v k L , ∞ ( uv ) with constant depending on the A constant of u and the A ∞ constant of v . This finishesthe proof of Corollary 1.12 after applying Theorem 1.2.4. Quantitative estimates
In the main theorem of this paper, Theorem 1.2, we show that the operator f → M ( fv ) v is bounded from L ( uv ) to L , ∞ ( uv ) and this bound depends on the A constant of u andthe A ∞ constant of v . For many reasons it would be very desirable to find a more precisebound. This is the purpose of this section, namely to quantify this bound.4.1. Dyadic maximal functions.
As in the proof of Theorem 1.2, we reduce mattersto the dyadic maximal function. We prove the following result.
Theorem 4.1.
Suppose that v ∈ A p and u ∈ A , where p > . Then (cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c n [ u ] A [ v ] A ∞ ([ u ] A [ v ] A ∞ + [ u ] A ∞ max { p, v ] A p + 1) } ) k f k L ( uv ) . Proof.
The proof is essentially given in the proof of Theorem 1.2, here we only track thedependence on the constant. Following the same notation, it is easy to check that X k ∈ Z X I kj,i ∈ Γ − ,k a k +1 u ( I kj,i ) ≤ c n [ u ] A [ v ] A ∞ k g k L ( u ) . For the remaining term, notice that, for any v ∈ A p , we have v ∈ A q for any q ≥ p ,moreover [ v ] A q ≤ [ v ] A p . So the same calculations give us | E k ∩ I kj || I kj | ≤ a − lq − [ v ] q − A q . Let q = max { p, v ] A p + 1) } , then[ v ] q − A q ≤ [ v ] v ] Ap +1) A p ≤ e. By Lemma 2.3, we obtain u ( E k ∩ I kj ) u ( I kj ) ≤ ea − lq − ) τn [ u ] A ∞ . Then following the same arguments we conclude that X k ∈ Z X l ≥ X I kj ∈ Γ l,k a k +1 u ( E k ∩ I kj ) ≤ c n max { p, v ] A p + 1) } [ u ] A [ u ] A ∞ [ v ] A ∞ k g k L ( u ) . (cid:3) ROOF OF AN EXTENSION OF SAWYER’S CONJECTURE 13
Remark 4.2.
We remark that in the special case of u = 1 , our arguments already give us X k ∈ Z X l ≥ X I kj ∈ Γ l,k a k +1 | E k ∩ I kj | ≤ c n max { p, v ] A p + 1) } [ v ] A ∞ k g k L ( u ) . For the remaining term, let K denote the maximal cubes in ∪ k ≥ N Γ − ,k . Then X k ∈ Z X I kj,i ∈ Γ − ,k a k +1 | I kj,i | ≤ a X k ∈ Z X I kj,i ∈ Γ − ,k v ( I kj,i ) ≤ c n [ v ] A ∞ X I ts,ℓ ∈K v ( I ts,ℓ ) ≤ c n [ v ] A ∞ X I ts,ℓ ∈K a h g i I ts | I ts,ℓ |≤ c ′ n [ v ] A ∞ k g k L ( R n ) , the last inequality holds since I ts are pairwise disjoint (due to the maximality of I ts,ℓ andLemma 2.2). So our technique recovers [OPR, Theorem 1.13] :Let v ∈ A p , p ≥ , then there exists a dimensional constant c such that (cid:13)(cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13)(cid:13) L , ∞ ( v ) ≤ c [ v ] A ∞ max { p, log( e + [ v ] A p ) }k f k L ( v ) . For the case of v ∈ A , there is a conjecture in [OPR], which states as follows Conjecture 4.3.
Let u, v ∈ A . Then there exists a dimensional constant c n such that (cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c n [ u ] A [ v ] A k f k L ( uv ) . In the following, we will give a quantitative bound which is far from the conjecture butstill improves the bound given in [OPR]. We also give a positive answer to the conjecturewhen u = v . Theorem 4.4.
Suppose that v ∈ A and u ∈ A . Then (cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c n [ u ] A [ v ] A ∞ ([ u ] A [ v ] A ∞ + log[ v ] A ) k f k L ( uv ) . Proof.
In the case of v ∈ A , we have a k + l < h v i I kj ≤ [ v ] A ess inf y ∈ I kj v ( y ) ≤ [ v ] A a k +1 . Then l ≤ c n (1 + log[ v ] A ) and the result follows. (cid:3) Theorem 4.5.
Suppose that u ∈ A , then (cid:13)(cid:13)(cid:13) M ( f u ) u (cid:13)(cid:13)(cid:13) L , ∞ ( u ) ≤ c n [ u ] A k f k L ( u ) . Proof.
The proof is still following the structure and notations of Theorem 1.2. First weconsider the case l ≥
0. Fix l , form the principal cubes for ∪ k ≥ N Γ l,k : let P l be themaximal cubes in ∪ k ≥ N Γ l,k , then for m ≥
0, if I ts ∈ P lm , we say I kj ∈ P lm +1 if I kj ismaximal in D ( I ts ) such that h u i I kj > h u i I ts Denote P l = ∪ m ≥ P lm and π ( Q ) is the minimal principal cube which contains Q . We have X k X I kj ∈ Γ l,k a k +1 u ( E k ∩ I kj ) ≤ a − l X k X I kj ∈ Γ l,k h u i I kj u ( E k ∩ I kj ) ≤ a − l X I ts ∈P l h u i I ts X k,j : π ( I kj )= I ts u ( E k ∩ I kj ) ≤ a − l X I ts ∈P l h u i I ts u ( I ts ) ≤ a X I ts ∈P l a t u ( I ts ) ≤ a Z R n g ( x ) X I ts ∈P l h u i I ts χ I ts ( x ) dx ≤ c n [ u ] A k g k L ( u ) . Finally, take the sum over 0 ≤ l ≤ c n (1 + log[ v ] A ) we obtain X k ∈ Z X l ≥ X I kj ∈ Γ l,k a k +1 u ( I kj ) ≤ c n [ u ] A (log[ u ] A + 1) k g k L ( u ) . It remains to treat the case l = −
1. In this case, we need to estimate X k ∈ Z X I kj,i ∈ Γ − ,k a k +1 u ( I kj,i ) =: X ( j,i,k ) ∈ Λ a k +1 u ( I kj,i ) . Keep in mind that in this case a k < h u i I kj,i ≤ a k +1 . Split the collection { I kj,i } ( j,i,k ) ∈ Λ := ∪ b Π b := ∪ b n I kj,i : 2 b ≤ h u i I kj,i h u i I kj < b +1 o . Since in this case, h u i I kj ≤ a k < h u i I kj,i , we know b ≥
0. Also notice that P i : I kj,i ∈ Π b | I kj,i || I kj | ≤ − b u ( ∪ i : I kj,i ∈ Π b I kj,i ) u ( I kj ) ≤ − b +1 (cid:16) P i : I kj,i ∈ Π b | I kj,i || I kj | (cid:17) τn [ u ] A ∞ , then P i : I kj,i ∈ Π b | I kj,i || I kj | ≤ − b (1+ τn [ u ] A ∞ ) . We also need the following observation. Namely, if I t i s i ,ℓ i ∈ Π b such that I t s ) I t s ) · · · ,then h u i I t s ≤ − b h u i I t s ,ℓ ≤ − b a t +1 − t m h u i I tmsm,ℓm < a t − t m h u i I tmsm . With all the above observations, we have X ( j,i,k ) ∈ Λ a k +1 u ( I kj,i ) ≤ a Z g ( x ) (cid:16) X ( j,i,k ) ∈ Λ u ( I kj,i ) | I kj | χ I kj ( x ) (cid:17) dx ROOF OF AN EXTENSION OF SAWYER’S CONJECTURE 15 ≤ a Z g ( x ) (cid:16) X b ≥ X I kj,i ∈ Π b u ( I kj,i ) | I kj | χ I kj ( x ) (cid:17) dx ≤ a Z g ( x ) (cid:16) X b ≥ b X I kj,i ∈ Π b h u i I kj | I kj,i || I kj | χ I kj ( x ) (cid:17) dx Now fix x , suppose I k m := I k m j m is the chain such that I k m ∋ x and there exists at leastone I k m j m ,i m ∈ Π b . Then X I kj,i ∈ Π b h u i I kj | I kj,i || I kj | χ I kj ( x ) = X m h u i I km X i m | I k m j m ,i m || I k m | ≤ − b (1+ τn [ u ] A ∞ ) X m h u i I km ≤ c n − b (1+ τn [ u ] A ∞ ) M u ( x ) . Finally, take the summation over b we conclude that X ( j,i,k ) ∈ Λ a k +1 u ( I kj,i ) ≤ c n [ u ] A [ u ] A ∞ k g k L ( u ) . (cid:3) One might be also interested in the quantitative bound of the case v ∈ A ∞ . To thisend, we need the following quantitative embedding result. Lemma 4.6. [HaPa]
Let w ∈ A ∞ . Then there exists dimensional constant c n such that w ∈ A p for p > e c n [ w ] A ∞ with [ w ] A p ≤ e e cn [ w ] A ∞ . Combining Theorem 4.1 and Lemma 4.6 we obtain the following
Corollary 4.7.
Suppose that v ∈ A ∞ and u ∈ A . Then (cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c n [ u ] A [ v ] A ∞ ([ u ] A [ v ] A ∞ + [ u ] A ∞ e c n [ v ] A ∞ ) k f k L ( uv ) . Calder´on-Zygmund operators.
In this section, we shall give a quantitative esti-mate of the following inequality (cid:13)(cid:13)(cid:13)(cid:13) T ( f v ) v (cid:13)(cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c u,v k f k L ( uv ) , where T is a Calder´on-Zygmund operator and u, v ∈ A . Essentially, the proof will followthe idea in [CMP2]. However, we will make slight changes to give a quantitative relationbetween the Calder´on-Zygmund operators and maximal operators. For u, v ∈ A , it iseasy to check that vu − p ∈ A p with[ vu − p ] A p ≤ [ v ] A [ u ] p − A . Define S ( f )( x ) = M ( f u )( x ) u ( x ) . Observe that k S ( f ) k L ∞ ( uv ) ≤ [ u ] A k f k L ∞ ( uv ) , and that k S ( f ) k L p, ( uv ) ≤ k S ( f ) k L p ( uv ) = k M ( f u ) k L p ( vu − p ) ≤ c n p ′ [ v ] p − A [ u ] A , where the last step is due to Buckley [B]. By interpolation (see e.g. [CMP2, PropositionA.1]), for p < q < ∞ , we obtain k S ( f ) k L q, ( uv ) ≤ q (cid:16) c n p ′ [ v ] p − A [ u ] A ( 1 p − q ) − + [ u ] A (cid:17) k f k L q, ( uv ) Let p = log([ v ] A + e ), then for any q ≥ u ] A log([ v ] A + e ) , one can check k S ( f ) k L q, ( uv ) ≤ c ′ n [ u ] A log([ v ] A + e ) k f k L q, ( uv ) . We denote K := c ′ n [ u ] A log([ v ] A + e )and we follow the Rubio de Francia algorithm: R h ( x ) := ∞ X k =0 S k ( h )( x )2 k K k . Easily we can check(1) h ( x ) ≤ R h ( x );(2) kR h k L r ′ , ( uv ) ≤ k h k L r ′ , ( uv ) ;(3) [( R h ) u ] A ≤ K .Here r is sharp reverse H¨older constant of ( R h ) u , equivalently, r ′ ≃ C n K by Lemma 2.3.Finally, following the argument in [CMP2], we have by duality of the Lorentz spaces andfor some parameter r > (cid:13)(cid:13)(cid:13) T ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) = (cid:13)(cid:13)(cid:13)(cid:16) T ( f v ) v (cid:17) r (cid:13)(cid:13)(cid:13) rL r, ∞ ( uv ) = sup h : k h k Lr ′ , uv ) =1 (cid:18)Z ( T ( f v )) r u ( x ) v ( x ) r ′ h ( x ) dx (cid:19) r ≤ sup h : k h k Lr ′ , uv ) =1 (cid:18)Z ( T ( f v )) r ( R h ) u ( x ) v ( x ) r ′ dx (cid:19) r Since ( R h ) u ∈ A and v ∈ A , we have1 | Q | Z Q ( R h ) u ( x ) v ( x ) r ′ ≤ (cid:16) | Q | Z Q (( R h ) u ) r (cid:17) r (cid:16) | Q | Z Q v (cid:17) r ′ ≤ (cid:16) | Q | Z Q ( R h ) u (cid:17)(cid:16) | Q | Z Q v (cid:17) r ′ ≤ K e inf x ∈ Q ( Rh )( x ) u ( x ) v ( x ) r ′ , and hence ( Rh ) uv r ′ ∈ A with a constant [( Rh ) uv r ′ ] A ≤ K . We use now a moreprecise Coifman-Fefferman estimate like (1.8) for T : let p ∈ (0 , ∞ ) and let w ∈ A ∞ , then k T f k L p ( w ) ≤ c p,T [ w ] A ∞ k M f k L p ( w ) ROOF OF AN EXTENSION OF SAWYER’S CONJECTURE 17 for any smooth function such that the left-hand side is finite (the proof in [LOP3] ofLemma 2.1 can be adapted to this situation). Then, since r >
1, we continue with Z ( T ( f v )) r ( R h ) u ( x ) v ( x ) r ′ dx ≤ c n,T K /r Z ( M ( f v )) r ( R h ) u ( x ) v ( x ) r ′ dx ≤ c n,T K (cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13) r L , ∞ ( uv ) kR h k L r ′ , ( uv ) ≤ c n,T K (cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13) r L , ∞ ( uv ) , where we have used H¨older’s inequality within the context of Lorentz spaces. Altogether,we obtain Theorem 4.8.
Let T be a Calder´on-Zygmund operator and u, v ∈ A . Then (cid:13)(cid:13)(cid:13) T ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) ≤ c n,T [ u ] A log([ v ] A + e ) (cid:13)(cid:13)(cid:13) M ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( uv ) . Combining Theorems 4.8 and 4.4, we obtain the following results, the first one recovers[LOP2, Theorem 1.4].
Corollary 4.9.
Let T be a Calder´on-Zygmund operator and v ∈ A . Then (cid:13)(cid:13)(cid:13) T ( f v ) v (cid:13)(cid:13)(cid:13) L , ∞ ( v ) ≤ c n,T [ v ] A log([ v ] A + e ) k f k L ( v ) . Corollary 4.10.
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ROOF OF AN EXTENSION OF SAWYER’S CONJECTURE 19 (K. Li)
BCAM, Basque Center for Applied Mathematics, Mazarredo, 14. 48009 BilbaoBasque Country, Spain
E-mail address : [email protected] (S. Ombrosi) Department of Mathematics, Universidad Nacional del Sur, Bah´ıa Blanca,Argentina
E-mail address : [email protected] (C. P´erez) Department of Mathematics, University of the Basque Country, Ikerbasqueand BCAM, Spain
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