Two-weight extrapolation on function spaces and applications
aa r X i v : . [ m a t h . C A ] J a n TWO-WEIGHT EXTRAPOLATION ON FUNCTION SPACES ANDAPPLICATIONS
MINGMING CAO AND ANDREA OLIVO
Abstract.
This paper is devoted to studying the two-weight extrapolation theory of Rubio deFrancia. We start to establish endpoint extrapolation results including A , A p and A ∞ extrap-olation in the context of Banach function spaces and modular spaces. Furthermore, we presentextrapolation for commutators in weighted Banach function spaces. Beyond that, we give sev-eral applications which can be easily obtained using extrapolation. First, we get local decayestimates for various operators and commutators. Second, Coifman-Fefferman inequalities areestablished and can be used to show some known sharp A inequalities. Third, Muckenhoupt-Wheeden and Sawyer conjectures are also presented for many operators, which go beyondCalder´on-Zygmund operators. Finally, we obtain two-weight inequalities for Littlewood-Paleyoperators and Fourier integral operators on weighted Banach function spaces. Contents
1. Introduction 22. Preliminaries 32.1. Muckenhoupt weights 32.2. Orlicz maximal operators 52.3. Banach function spaces 83. Extrapolation on Banach function spaces 143.1. From one-weight to two-weight 143.2. From two-weight to two-weight 234. Extrapolation on modular spaces 275. Extrapolation for commutators 326. Applications 386.1. Local decay estimates 396.2. Coifman-Fefferman inequalities 426.3. Muckenhoupt-Wheeden conjecture 43
Date : January 16, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Rubio de Francia extrapolation, Banach function spaces, Modular spaces, Two-weight inequalities, Local decay estimates, Muckenhoupt-Wheeden conjecture, Vector-valued inequalities.The first author acknowledges financial support from the Spanish Ministry of Science and Innovation, throughthe Juan de la Cierva-Formaci´on 2018 (FJC2018-038526-I), through the “Severo Ochoa Programme for Centres ofExcellence in R&D” (CEX2019-000904-S) and from the Spanish National Research Council, through the “Ayudaextraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001). The second author was supported byGrants UBACyT 20020170100430BA (University of Buenos Aires), PIP 11220150100355 (CONICET) and PICT2018-03399.
Introduction
One of the most useful and powerful tools in harmonic analysis is the Rubio de Franciaextrapolation theorem [65], which states that if an operator T is bounded on L p ( w ) for some p ∈ [1 , ∞ ) and for all w ∈ A p , then T is bounded in L p ( w ) for all p ∈ (1 , ∞ ) and for all w ∈ A p . Over the years, Rubio de Francia’s result has been extended and complemented indifferent manners, see [21] and the references therein. These results only involve the L p ( w )boundedness of an operator T , which can be generally formulated by the pair of functions.In addition to the extrapolation theorems aforementioned, there are several other kinds ofextrapolation. For example, B´enyi et al. [10] obtained comprehensive extrapolation resultsfrom T to its commutator [ b, T ] on weighted Lebesgue spaces. Recently, Hyt¨onen and Lappas[40, 41] established a “compact version” of Rubio de Francia’s extrapolation theorem, whichallows one to extrapolate the compactness of an operator from just one space to the full rangeof weighted spaces, provided that the operator is bounded. This result has been extended tothe multilinear case in [13]. Another kind of extrapolation, related with second order ellipticoperators, was presented by Shen [67], where it was proved that one can extrapolate thesolvability of the L p -Dirichlet problem on Lipschitz domains from on some exponent p toa large range of p ’s. Beyond that, Hofmann and Martell [36, 37] built the extrapolation ofCarleson measure in order to investigate the uniform rectifiability and A ∞ property of ellipticmeasures. To sum up, a common point in all extrapolation theorems is that one can obtainthe global or local information from an estimate at certain single point.The Rubio de Francia extrapolation theorem and its variants have been proved to be ex-tremely advantageous and the key to solving many problems in harmonic analysis. Indeed,extrapolation theorems allow us to reduce the general L p estimates for certain operators to asuitable case p = p , for example, see [14] for the Coifman-Fefferman’s inequality for p = 1,[39] for the Calder´on-Zygmund operators for p = 2, and [47] for square functions for p = 3.Even more, the technique of extrapolation can refine some estimates, see [20] for the Sawyerconjecture, [48, 49] for the weak Muckenhoupt-Wheeden conjecture and [61] for the local decayestimates. Also, using extrapolation theorems, one can obtain sharp weighted inequalities forseveral operators, see [10] and [45]. Another interesting point is that by means of extrapola-tion, the vector-valued inequalities immediately follows from the corresponding scalar-valuedinequalities, see Theorems 3.4, 3.11, 3.15 and 4.1 below.Recently, Mar´ın, Martell and the first author [12] generalized the Rubio de Francia extrap-olation theory to the context of Banach Function spaces (BFS) over measure spaces. To dothis, they formulated the weighted boundedness of Hardy-Littlewood maximal operators onweighted BFS, which is the natural substitute of Muckenhoupt weights in this general setting.Motivated by these results, we are going to establish the two-weight extrapolation on BFS over( R n , dx ) (cf. Definition 2.8), which includes Lorentz spaces, variable Lebesgue spaces, Orliczspaces, etc.In this paper we present a collection of results about two-weight extrapolation theoremson BFS as well as numerous applications of them. First, Section 3 is devoted to obtaining WO-WEIGHT EXTRAPOLATION 3 two-weight extrapolation theorems in the context of BFS over ( R n , dx ). Our first theorem isan endpoint A p extrapolation (cf. Theorem 3.1), which can be used in Section 6.1 to establishlocal decay estimates for various operators by the means of sparse domination. Such type ofestimates originated in [44] and were further investigated in [61]. After that, we present an A and a A ∞ extrapolation result (cf. Theorems 3.3 and 3.4), which should be comparedto [12, Theorems 3.3 and 3.5]. Furthermore, we include several A ∞ extrapolation theoremswhich can be applied to some well-known problems. Both Theorem 3.5 and Theorem 3.6 refine[19, Theorem 2.1] in the weighted BFS. Additionally, by means of Theorems 3.6 and 3.7, weobtain two kinds of Coifman-Fefferman inequalities in BFS (cf. Section 6.2). Theorem 3.8allows us to formulate Sawyer conjecture for singular integrals on the general Banach functionspaces and Theorem 3.10 leads us to present some estimates concerning to the dual version ofMuckenhoupt-Wheeden conjecture.In [23], Cruz-Uribe and P´erez obtained several extrapolations result for pairs of weights of theform ( w, M k w ) or ( w, ( M w/w ) r w ), where r > M k is the k th iteration of M . Continuingwith this line, we prove Theorems 3.11 and 3.12 respectively extending [23, Theorem 1.3] and[21, Theorem 8.2] to the weighted BFS. On the other hand, with Theorem 3.14 and Theorem3.15, we recover the sharp vector-valued Fefferman-Stein inequality obtained in [23, Theorem1.4] and [63, Theorem 1.1]. Then we also proved two-weight inequalities for certain Littlewood-Paley operators (cf. Section 6.4) and Fourier integral operators (cf. Section 6.5).In Section 4, for a Young function Φ and a weight w , we give two-weight extrapolation resultson modular spaces ρ Φ w . Theorem 4.1 can be viewed as a two-weight version of [24, Theorem3.1]. Then Corollary 4.2 is a hybrid of Theorem 3.4 and Theorem 4.1. As a consequence,Corollary 4.3 gives a control of an operator T by the maximal operator M A locally. In addition,compared to [12, Theorems 4.6 and 4.7], Theorems 4.4 and 4.5 are respectively formulated bythe boundedness of M v instead of M ′ v . Based on these results above, in Section 6.2, we obtainCoifman-Fefferman type’s inequalities on modular spaces, recovering those in [16, 46, 54].We end up with two-weight extrapolation for commutators in Section 5. As we mentionpreviously, the authors in [10] obtained extrapolation results for commutators on weightedLebesgue spaces. Following this idea, we establish a sharp two-weight analogue in the set-ting of weighted Banach function spaces (cf. Theorem 5.1). Then Bloom type estimates forcommutators originated in [11] are given in Theorems 5.3 and 5.4.2. Preliminaries
Muckenhoupt weights.
We briefly recall the notion of Muckenhoupt weights and therelevant properties in this section. Given a locally integrable function f on R n , the Hardy-Littlewood maximal operator M is defined by M f ( x ) := sup Q ∋ x Q | f ( y ) | dy, x ∈ R n , (2.1)where the supremum is taken over all cubes Q ⊂ R n containing x .We say that a measurable function w defined on R n is a weight if 0 < w ( x ) < ∞ a.e. x ∈ R n . Given p ∈ (1 , ∞ ), we define the Muckenhoupt class A p as the collection of allweights w satisfying[ w ] A p := sup Q ⊂ R n (cid:18) Q w ( x ) dx (cid:19) (cid:18) Q w ( x ) − p ′ dx (cid:19) p − < ∞ , (2.2) MINGMING CAO AND ANDREA OLIVO where the supremum is taken over all cubes Q ⊂ R n and p ′ is the H¨older conjugate exponentof p , i.e., 1 /p + 1 /p ′ = 1. In the case p = 1, we say that w ∈ A if[ w ] A := k M w/w k L ∞ ( R n ) < ∞ . Then, we define A ∞ = [ p ≥ A p . For every p ∈ (1 , ∞ ) and every weight w , we define the associated weighted Lebesgue space L p ( w ) := L p ( R n , w dx ) as the set of measurable functions f with ´ R n | f | p w dx < ∞ .Let 1 ≤ p ≤ q < ∞ . We say that w ∈ A p,q if it satisfies[ w ] A p,q := sup Q (cid:18) Q w q dx (cid:19) (cid:18) Q w − p ′ dx (cid:19) qp ′ < ∞ . Observe that w ∈ A p,q if and only if w q ∈ A qp ′ if and only if w − p ′ ∈ A p ′ q . Moreover,[ w ] A p,q = [ w q ] A qp ′ = [ w − p ′ ] qp ′ A p ′ q . (2.3)Together with p ≤ q , the equation (2.3) implies that w ∈ A p,q ⇐⇒ w p ∈ A p and w q ∈ A q . (2.4)For s ∈ (1 , ∞ ], we define the reverse H¨older class RH s as the collection of all weights w such that [ w ] RH s := sup Q (cid:18) Q w s dx (cid:19) s (cid:18) Q w dx (cid:19) − < ∞ . When s = ∞ , ( ffl Q w s dx ) /s is understood as (ess sup Q w ).Given p ∈ (1 , ∞ ) and a weight w , the Muckenhoupt’s theorem [57] states that the Hardy-Littlewood maximal operator M is bounded on L p ( w ) if and only if w ∈ A p . More precisely,there is a constant C = C ( n, p ) such that k M k L p ( w ) → L p ( w ) ≤ C [ w ] p − A p . (2.5)The properties of Muckenhoupt weights are well-known (cf., e.g [31]). We mention some ofthem here for completeness.(a) For all 1 ≤ p ≤ q ≤ ∞ , A p ⊆ A q with [ w ] A q ≤ [ w ] A p , ∀ w ∈ A p .(b) Let w ∈ A p with p ∈ (1 , ∞ ). Then there are ε ∈ (0 , p −
1) and τ ∈ (1 , ∞ ) such that w ∈ A p − ε and w τ ∈ A p .(c) For every p ∈ (1 , ∞ ), w ∈ A p if and only if w − p ′ ∈ A p ′ , and [ w − p ′ ] A p ′ = [ w ] p ′ − A p .(d) For every p ∈ (1 , ∞ ) and w , w ∈ A , w w − p ∈ A p with [ w w − p ] A p ≤ [ w ] A [ w ] p − A . (2.6)(e) w ∈ A ∞ if and only if w ∈ RH s for some s ∈ (1 , ∞ ). WO-WEIGHT EXTRAPOLATION 5 (f) For any positive Borel measure µ ,[( M µ ) δ ] ≤ c n − δ , ∀ δ ∈ (0 , . (2.7)As a consequence, 1 ≤ [( M µ ) − λ ] RH ∞ ≤ c n,λ , ∀ λ > . (2.8)By w ∈ A p ( u ), we mean that w satisfies the A p condition defined with respect to the measure u dx . The properties below considering the endpoint case were given in [20]. Lemma 2.1.
The following statements hold: (1) If u ∈ A , then u − ∈ RH ∞ . (2) If u ∈ A ∞ and v ∈ RH ∞ , then uv ∈ A ∞ . (3) If u ∈ RH ∞ , then u s ∈ RH ∞ for any s > . (4) u ∈ A ∞ if and only if u = u u , where u ∈ A and u ∈ RH ∞ . (5) If u ∈ A and v ∈ A p ( u ) with ≤ p < ∞ , then uv ∈ A p with [ uv ] A p ≤ [ u ] pA [ v ] A p ( u ) . (6) If u ∈ A p with ≤ p ≤ ∞ and v ∈ A ( u ) , then uv ∈ A p with [ uv ] A p ≤ [ u ] A p [ v ] A ( u ) . (7) If u ∈ A and v ∈ A p with ≤ p < ∞ , then there exists < ǫ < depending only on [ u ] A such that uv ǫ ∈ A p for all < ǫ < ǫ . Orlicz maximal operators.
A function Φ : [0 , ∞ ) → [0 , ∞ ) is called a Young function if it is continuous, convex, strictly increasing, and satisfieslim t → + Φ( t ) t = 0 and lim t →∞ Φ( t ) t = ∞ . Given p ∈ [1 , ∞ ), we say that a Young function Φ is a p -Young function , if Ψ( t ) = Φ( t /p ) isa Young function.If A and B are Young functions, we write A ( t ) ≃ B ( t ) if there are constants c , c > c A ( t ) ≤ B ( t ) ≤ c A ( t ) for all t ≥ t >
0. Also, we denote A ( t ) (cid:22) B ( t ) if there exists c > A ( t ) ≤ B ( ct ) for all t ≥ t >
0. Note that for all Young functions φ , t (cid:22) φ ( t ).Further, if A ( t ) ≤ cB ( t ) for some c >
1, then by convexity, A ( t ) ≤ B ( ct ).A function Φ is said to be doubling , or Φ ∈ ∆ , if there exists a constant C > t ) ≤ C Φ( t ) for any t >
0. Given a Young function Φ, its complementary function¯Φ : [0 , ∞ ) → [0 , ∞ ) is defined by¯Φ( t ) := sup s> { st − Φ( s ) } , t > , which clearly implies that st ≤ Φ( s ) + ¯Φ( t ) , s, t > . (2.9)Moreover, one can check that ¯Φ is also a Young function and t ≤ Φ − ( t ) ¯Φ − ( t ) ≤ t, t > . (2.10)In turn, by replacing t by Φ( t ) in first inequality of (2.10), we obtain¯Φ (cid:16) Φ( t ) t (cid:17) ≤ Φ( t ) , t > . (2.11) MINGMING CAO AND ANDREA OLIVO
Let us recall the lower and upper dilation indices of a positive increasing function Φ on[0 , ∞ ), which are respectively defined by i Φ := lim t → + log h Φ ( t )log t and I Φ := lim t →∞ log h Φ ( t )log t , (2.12)where h Φ is defined as h Φ ( t ) := sup s> Φ( st )Φ( s ) , t > . From the definitions, one can show that1 ≤ i Φ ≤ I Φ ≤ ∞ , ( I Φ ) ′ = i ¯Φ , and ( i Φ ) ′ = I ¯Φ . Furthermore, it turns out that Φ ∈ ∆ if and only if I Φ < ∞ , and henceΦ , ¯Φ ∈ ∆ if and only if 1 < i Φ ≤ I Φ < ∞ . (2.13)We conclude by giving some examples of the lower and upper dilation indices. • Let Φ( t ) = t p , 1 < p < ∞ . Then ¯Φ( t ) = pt p ′ /p ′ and i Φ = I Φ = p . • Let Φ( t ) ≃ t p log( e + t ) α with 1 < p < ∞ and α ∈ R . Then ¯Φ( t ) ≃ t p ′ log( e + t ) α (1 − p ′ ) and i Φ = I Φ = p . • Given 1 < p < ∞ , let Φ( t ) ≃ t p , 0 ≤ t ≤
1, and Φ( t ) ≃ e t , t ≥
1. Then ¯Φ( t ) ≃ t p ′ ,0 ≤ t ≤
1, and ¯Φ( t ) ≃ t log( e + t ), t ≥
1. In this case, i Φ = p and I Φ = ∞ , and Φ ∆ .Given a Young function Φ, we define the Orlicz space L Φ (Ω , µ ) to be the function spacewith Luxemburg norm k f k L Φ (Ω ,µ ) := inf (cid:26) λ > ˆ Ω Φ (cid:16) | f ( x ) | λ (cid:17) dµ ( x ) ≤ (cid:27) . (2.14)Now we define the Orlicz maximal operator M Φ f ( x ) := sup Q ∋ x k f k Φ ,Q := sup Q ∋ x k f k L Φ ( Q, dx | Q | ) , where the supremum is taken over all cubes Q in R n . When Φ( t ) = t p , 1 ≤ p < ∞ , k f k Φ ,Q = (cid:18) Q | f ( x ) | p dx (cid:19) p =: k f k p,Q . In this case, if p = 1, M Φ agrees with the classical maximal operator M in (2.1); if p > M Φ f = M p f := M ( | f | p ) /p . If Φ( t ) (cid:22) Ψ( t ), then M Φ f ( x ) ≤ cM Ψ f ( x ) for all x ∈ R n .The classical H¨older’s inequality can be generalized to Orlicz spaces [59]. Lemma 2.2.
Given a Young function A , then for all cubes Q , Q | f g | dx ≤ k f k A,Q k g k ¯ A,Q . (2.15) More generally, if A , B and C are Young functions such that A − ( t ) B − ( t ) ≤ c C − ( t ) , forall t ≥ t > , then k f g k C,Q ≤ c k f k A,Q k g k B,Q . (2.16)The following result is an extension of the well-known Coifman-Rochberg theorem (2.7).The proof can be found in [43, Lemma 4.2]. WO-WEIGHT EXTRAPOLATION 7
Lemma 2.3.
Let Φ be a Young function and w be a nonnegative function such that M Φ w ( x ) < ∞ a.e.. Then [( M Φ w ) δ ] A ≤ c n,δ , ∀ δ ∈ (0 , , (2.17)[( M Φ w ) − λ ] RH ∞ ≤ c n,λ , ∀ λ > . (2.18)Given p ∈ (1 , ∞ ), a Young function Φ is said to satisfy the B p condition (or, Φ ∈ B p ) iffor some c > ˆ ∞ c Φ( t ) t p dtt < ∞ . (2.19)Observe that if (2.19) is finite for some c >
0, then it is finite for every c >
0. Let [Φ] B p denotethe value if c = 1 in (2.19). It was shown in [21, Proposition 5.10] that if Φ and ¯Φ are doublingYoung functions, then Φ ∈ B p if and only if ˆ ∞ c (cid:18) t p ′ ¯Φ( t ) (cid:19) p − dtt < ∞ . Let us present two types of B p bump conditions. An important special case is the “log-bumps” of the form A ( t ) = t p log( e + t ) p − δ , B ( t ) = t p ′ log( e + t ) p ′ − δ , δ > . (2.20)Another interesting example is the “loglog-bumps” as follows: A ( t ) = t p log( e + t ) p − log log( e e + t ) p − δ , δ > B ( t ) = t p ′ log( e + t ) p ′ − log log( e e + t ) p ′ − δ , δ > . (2.22)Then one can verify that in both cases above, ¯ A ∈ B p ′ and ¯ B ∈ B p for any 1 < p < ∞ .The B p condition can be also characterized by the boundedness of the Orlicz maximaloperator M Φ . Indeed, the following result was given in [21, Theorem 5.13] and [43, eq. (25)]. Lemma 2.4.
Let < p < ∞ . Then M Φ is bounded on L p ( R n ) if and only if Φ ∈ B p .Moreover, k M Φ k L p ( R n ) → L p ( R n ) ≤ C n,p [Φ] p B p . In particular, if the Young function A is the sameas the first one in (2.20) or (2.21) , then k M ¯ A k L p ′ ( R n ) → L p ′ ( R n ) ≤ c n p δ − p ′ , ∀ δ ∈ (0 , . (2.23) Definition 2.5.
Given p ∈ (1 , ∞ ) , let A and B be Young functions such that ¯ A ∈ B p ′ and ¯ B ∈ B p . We say that the pair of weights ( u, v ) satisfies the double bump condition withrespect to A and B if [ u, v ] A,B,p := sup Q k u p k A,Q k v − p k B,Q < ∞ . (2.24) where the supremum is taken over all cubes Q in R n . Also, ( u, v ) is said to satisfy the separated bump condition if [ u, v ] A,p ′ := sup Q k u p k A,Q k v − p k p ′ ,Q < ∞ , (2.25)[ u, v ] p,B := sup Q k u p k p,Q k v − p k B,Q < ∞ . (2.26) MINGMING CAO AND ANDREA OLIVO
Note that if A ( t ) = t p in (2.25) or B ( t ) = t p in (2.26), each of them actually is two-weight A p condition and we denote them by [ u, v ] A p := [ u, v ] p,p ′ . Also, the separated bump conditionis weaker than the double bump condition. Indeed, (2.24) implies (2.25) and (2.26), but thereverse direction is incorrect. The first fact holds since ¯ A ∈ B p ′ and ¯ B ∈ B p respectivelyindicate A is a p -Young function and B is a p ′ -Young function. The second fact was shown in[3, Section 7] by constructing log-bumps. Lemma 2.6.
Let < p < ∞ , let A , B and Φ be Young functions such that A ∈ B p and A − ( t ) B − ( t ) . Φ − ( t ) for any t > t > . If a pair of weights ( u, v ) satisfies [ u, v ] p,B < ∞ ,then k M Φ f k L p ( u ) ≤ C [ u, v ] p,B [ A ] p B p k f k L p ( v ) . (2.27) Moreover, (2.27) holds for Φ( t ) = t and B = ¯ A satisfying the same hypotheses. In this case, ¯ A ∈ B p is necessary. The two-weight inequality above was established in [21, Theorem 5.14] and [22, Theorem 3.1].The weak type inequality for M Φ was also obtained in [21, Proposition 5.16] as follows. Lemma 2.7.
Let < p < ∞ , let B and Φ be Young functions such that t p B − ( t ) . Φ − ( t ) for any t > t > . If a pair of weights ( u, v ) satisfies [ u, v ] p,B < ∞ , then k M Φ f k L p, ∞ ( u ) ≤ C k f k L p ( v ) . (2.28) Moreover, (2.28) holds for M if and only if [ u, v ] A p < ∞ . Banach function spaces.
Let M denote the collection of all (equivalence classes of)Lebesgue measurable functions f : R n → C . The characteristic function of the set E will bedenoted by E . Definition 2.8.
We say that a mapping k·k : M → [0 , ∞ ] is a function norm if it satisfiesthe following: (1) k f k = k| f |k and k f k = 0 if and only if f = 0 a.e. (2) k f + g k ≤ k f k + k g k . (3) k λf k = | λ | k f k for every λ ∈ R . (4) If | f | ≤ | g | a.e., then k f k ≤ k g k . (5) If { f j } j ∈ N ⊆ M is a sequence such that | f j | increases to | f | µ -a.e. as j → ∞ , then k f j k increases to k f k as j → ∞ . (6) If E ⊆ R n is a measurable set with | E | < ∞ , then k E k < ∞ and there is a constant C E such that ´ E | f | dx ≤ C E k f k . Given a function norm k · k , the set X = { f ∈ M : k f k < + ∞} (2.29)is called a Banach function space (BFS, for short) over ( R n , dx ). In such a scenario, we shallwrite k·k X in place of k·k in order to emphasize the connection between the function norm k·k and its associated function space X . Then ( X , k·k X ) is a Banach space.For a Banach function space X over ( R n , dx ), one can define its associate space X ′ ac-cording to X ′ = { f ∈ M : k f k X ′ < + ∞} , WO-WEIGHT EXTRAPOLATION 9 where k f k X ′ = sup (cid:26) ˆ R n | f ( x ) g ( x ) | dx : g ∈ X , k g k X ≤ (cid:27) , and with this definition X ′ is also a Banach function space.It follows from the definition of X ′ that the following generalized H¨older’s inequality holds: ˆ R n | f ( x ) g ( x ) | dx ≤ k f k X k g k X ′ , f ∈ X and g ∈ X ′ . (2.30)It turns out that X = ( X ′ ) ′ =: X ′′ (cf. [9, Theorem 2.7, p. 10]). Therefore, one has k f k X = sup (cid:26) ˆ R n | f ( x ) g ( x ) | dx : g ∈ X ′ , k g k X ′ ≤ (cid:27) . (2.31) Remark 2.9.
It is useful to note that the supremum in (2.31) does not change if it is takenonly over functions g ∈ X ′ with k g k X ′ ≤ which are non-negative and positive on a set ofpositive measure (that is, non-negative g ∈ X ′ with < k g k X ′ ≤ ). Indeed, the fact that wecan consider only non-negative functions is direct from (2.31) . If k f k X > then there is g ∈ X ′ with k g k X ′ ≤ such that < k f k X ≤ ´ R n | f g | dx and this forces g to be non-zero on a set ofpositive measure. Finally, the case k f k X = 0 is trivial. Given p ∈ (0 , ∞ ) and a Banach function space X , we define the scale of space X p by X p := { f ∈ M : | f | p ∈ X } with k f k X p := k| f | p k /p X , (2.32)which indicates that X p is also a Banach function space whenever p ∈ [1 , ∞ ).For every function f ∈ M , define its distribution function as µ f ( λ ) = |{ x ∈ R n : | f ( x ) | > λ }| , λ > . (2.33)A Banach function space X is called rearrangement invariant (RI, for short) if k f k X = k g k X for every pair of functions f, g ∈ X such that µ f = µ g . In particular, if X is a RIBFS,one can check that its associate space X ′ is also a RIBFS.For each f ∈ M , the decreasing rearrangement of f with respect to the Lebesgue measurein R n is the function f ∗ defined by f ∗ ( t ) = inf { λ ≥ µ f ( λ ) ≤ t } , t ∈ [0 , ∞ ) . (2.34)Note that the functions f and f ∗ have the same distribution function. One remarkable con-sequence of this is Luxemburg representation theorem: if X is a RIBFS then there exists aRIBFS X over [0 , ∞ ) such that f ∈ X if and only if f ∗ ∈ X and k f ∗ k X = k f k X (cf. [9, Theorem4.10, p. 62]). Using this representation we can define Boyd indices of a RIBFS X . Given f ∈ X ,consider the dilatation operator D t , 0 < t < ∞ , by setting D t f ( s ) := f ( s/t ) for each s ≥ h X ( t ) := sup (cid:8) k D t f k X : f ∈ X with k f k X ≤ (cid:9) , t > , (2.35)the lower and upper Boyd indices may, respectively, defined as p X := lim t →∞ log t log h X ( t ) = sup
Let < p < q < ∞ and w be a weight on R n . Suppose that X is a RIBFS over ( R n , dx ) with p < p X ≤ q X < q . If a sublinear operator T is bounded on L p ( w ) and L q ( w ) ,then T is bounded on X w . Let us see the boundedness of the Hardy-Littlewood maximal operator on these weightedRI space. Let M w be the weighted maximal operator defined by M w f ( x ) := sup Q ∋ x w ( Q ) ˆ Q | f ( y ) | w ( y ) dy, (2.39)where the supremum is taken over all cubes Q containing x . Moreover, we define M cw asthe weighted centered maximal operator if the supremum in (2.39) is taken over all cubes Q centered at x . Lemma 2.11.
Let X be a RIBFS over ( R n , dx ) with p X > . Then there exists a constant C such that the following hold: (a) M is bounded on X if and only if p X > . (b) M is bounded on X ( w ) for every w ∈ A p X . (c) For every weight w , M cw is bounded on X ( w ) with k M cw k X ( w ) → X ( w ) ≤ C . (d) For every w ∈ A ∞ , then M w is bounded on X ( w ) with k M w k X ( w ) → X ( w ) ≤ C (cid:16) Q ⊂ R n w (3 Q ) w ( Q ) (cid:17) . (2.40) WO-WEIGHT EXTRAPOLATION 11
Proof.
Note that (a) and (b) were respectively obtained in [56] and [21, Lemma 4.12]. To show(c) and (d), we will follow the strategy in [24] to pursue the accurate bound. It is shown in[32, p.509] that for every weight w , k M cw k L ( w ) → L , ∞ ( w ) ≤ c n , (2.41)where c n is independent of w . Invoking [4, Theorem 2], we obtain that M cw is of weak-type (1 ,
1) with respect to the measure w if and only if ( M cw f ) ∗ w ( t ) ≤ Cf ∗∗ w ( t ), t >
0, where f ∗∗ w ( t ) = t ´ t f ∗ w ( s ) ds . Moreover, a careful checking of the proof yields that k M cw k L ( w ) → L , ∞ ( w ) f ∗∗ w ( t ) ≤ ( M cw f ) ∗ w ( t ) ≤ (1 + k M cw k L ( w ) → L , ∞ ( w ) ) f ∗∗ w ( t ) , (2.42)for all t >
0. Combining (2.41) with (2.42), one has k M cw f k X ( w ) = k ( M cw f ) ∗ w k X ≤ (1 + c n ) k f ∗∗ w k X . (2.43)Also, observe that f ∗∗ w is the Hardy operator acting over f ∗ w . A conclusion from [56] assertsthat the Hardy operator is bounded on X if and only if p X > X is a RIBFS. Thus,together with (2.43), this implies k M cw f k X ( w ) ≤ C (1 + c n ) k f ∗ w k X = C n k f k X ( w ) . This shows (c). The proof of (d) is almost the same, but (2.41) is replaced by k M w k L ( w ) → L , ∞ ( w ) ≤ sup Q ⊂ R n w (3 Q ) w ( Q ) , ∀ w ∈ A ∞ , which can be found in [32, Exercise 2.1.1] since w ∈ A ∞ implies that wdx is a doublingmeasure. (cid:3) Finally, we present several examples of Banach function spaces.
Example 2.12 (Classical Lorentz spaces) . Let < p, q ≤ ∞ . The Lorentz spaces L p,q ( R n ) consist of all measurable functions f on R n with the quasi-norm k f k L p,q ( R n ) < ∞ , where k f k L p,q ( R n ) := (cid:18) ˆ ∞ (cid:0) t p f ∗ ( t ) (cid:1) q dtt (cid:19) q , q < ∞ , sup t> t p f ∗ ( t ) , q = ∞ . Observe that L p,p ( R n ) is the Lebesgue space L p ( R n ) for any < p ≤ ∞ .Let X = L p,q ( R n ) . Then X is RIBFS and p X = q X = p for all < p < ∞ and ≤ q ≤∞ , or p = q = ∞ (see [9, Theorem 4.6, p. 219] ). The associated space X ′ is given in [15,Example 2.4.43] by • X ′ = L p ′ ,q ′ ( R n ) , if < p, q < ∞ . • X ′ = L ∞ ( R n ) , if < q ≤ p = 1 . • X ′ = L p ′ , ∞ ( R n ) , if < q ≤ < p < ∞ . • X ′ = { } , if < p < and < q ≤ ∞ , or p < q < ∞ . Example 2.13 (Lorentz spaces Λ p,q ( w )) . Let w be a weight on R + , and let < p, q ≤ ∞ . TheLorentz space Λ p,q ( w ) is defined by Λ p,q ( w ) := { f ∈ M : k f k Λ p,q ( w ) := k f ∗ k L p,q ( w ) < ∞} . In this sequel, we write Λ p ( w ) := Λ p,p ( w ) for < p ≤ ∞ , and b w ( t ) := ´ t w ( s ) ds for everyweight w on R + . • If w ≡ , then Λ p,q ( w ) = L p,q ( R n ) for < p, q ≤ ∞ . • If w ( t ) = t qp − , then Λ q ( w ) = L p,q ( R n ) for < p, q < ∞ . • If w ( t ) = t qp − (1 + log + 1 t ) α , then Λ q ( w ) = L p,q (log L ) α is the Lorentz-Zygmund space,where < p, q < ∞ and α ∈ R , see [8] . • If w ( t ) = t qp − (1 + log + 1 t ) α (1 + log + log + 1 t ) β , then Λ q ( w ) = L p,q (log L ) α (log log L ) β isthe generalized Lorentz-Zygmund space, where < p, q < ∞ and α, β ∈ R . In this case, Λ p ( w ) is a BFS and Λ p ( w ) ′ = L p ′ (log L ) − α (log log L ) − β for < p < ∞ and α, β ∈ R , see [27] . • If w = (0 , , the space Λ ( w ) contains L ∞ ( R n ) . The functional k · k Λ ( w ) is a norm andthe space is a BFS.We collect some facts from [15, Section 2] . If w is decreasing or w ∈ B p, ∞ , then Λ p ( w ) isa BFS for ≤ p < ∞ . In addition, if b w ∈ ∆ , then Λ p,q ( w ) ′ is RIBFS for < p < ∞ and < q ≤ ∞ . For the associated spaces, it was proved that Λ p ( w ) ′ = Λ p ′ ( w − p ′ ) for < p < ∞ if w is increasing or w satisfies t ´ t w ( s ) ds ≤ Cw ( t ) for all t > .Next let us see the equivalence between Boyd indices and the boundedness of M . The theLorentz-Shimogaki theorem [56] says that for any < p < ∞ , M : Λ p ( w ) → Λ p ( w ) ⇐⇒ I b w < p, where I b w is given in (2.12) . If we set X = Λ p ( w ) with < p < ∞ , then the weighted versionof X is X ( u ) = Λ pu ( w ) for any weight u on R n , where Λ pu ( w ) := { f ∈ M : k f k Λ pu ( w ) := k f ∗ u k L p ( w ) < ∞} . Although X ( u ) may not be a BFS, Lerner and P´erez [52] obtained that M : X ( u ) → X ( u ) ⇐⇒ p X ( u ) > . (2.44) In particular, if we take w ≡ , then X ( u ) = Λ pu ( w ) = L p ( w ) . Then by (2.5) and (2.44) , wededuce that for every < p < ∞ , M : L p ( u ) → L p ( u ) ⇐⇒ u ∈ A p ⇐⇒ p L p ( u ) > . Example 2.14 (Grand Lebesgue spaces) . Let T = (0 , and w be a weight on T . Let M bethe set of all Lebesgue measurable real valued functions on T . Suppose that < δ ( · ) ∈ L ∞ ( T ) with k δ k L ∞ ( T ) ≤ , and p ( · ) ∈ M with p ( · ) ≥ a.e.. Denote L p [ · ] ,δ ( · ) ( T , w ) := { f ∈ M : ρ p [ · ] ,δ ( · ) ,w ( f ) < ∞} , where ρ p [ · ] ,δ ( · ) ,w ( f ) = ess sup x ∈ T (cid:18) ˆ T | δ ( x ) f ( t ) | p ( x ) w ( t ) dt (cid:19) p ( x ) . The collection of functions L p [ · ] ,δ ( · ) ( T , w ) is called the weighted fully measurable grandLebesgue space . It was proved in [2, Proposition 2] that L p [ · ] ,δ ( · ) ( T , w ) is a BFS for every weight w on T . (2.45) Moreover, M T : L p [ · ] ,δ ( · ) ( T , w ) → L p [ · ] ,δ ( · ) ( T , w ) if and only if w ∈ A p + ( T ) , (2.46) where M T is the Hardy-Littlewood maximal operator restricted on T . WO-WEIGHT EXTRAPOLATION 13 If p ( · ) ≡ p with ≤ p < ∞ and δ ( · ) ≡ , the space L p [ · ] ,δ ( · ) ( T , w ) coincides with the weightedLebesgue space L p ( T , w ) . Let L p ) ,δ ( · ) ( T ) denote L p [ · ] ,δ ( · ) ( T , w ) if p ( x ) = p − x , δ ( x ) = η ( x ) p − x is increasing and w ≡ . Let L p ) ( T , w ) denote L p [ · ] ,δ ( · ) ( T , w ) if p ( x ) = p − x , δ ( x ) = x p − x . Theresults (2.45) and (2.46) contains the particular cases in [29] and [30] . Beyond that, Formicaand Giova [30] obtained the Boyd indices: X := L p ) ,δ ( · ) ( T ) is a RIBFS and p X = q X = p. Example 2.15 (Musielak-Orlicz space L ϕ ( · ) ( R n )) . A convex, left-continuous function φ :[0 , ∞ ) → [0 , ∞ ] with lim t → + φ ( t ) = φ (0) = 0 , and lim t →∞ φ ( t ) = ∞ is called a Φ -function. Let Ξ( R n ) be the collection of functions ϕ : R n × [0 , ∞ ) → [0 , ∞ ] such that ϕ ( y, · ) is a Φ -functionfor every y ∈ R n , and ϕ ( · , t ) ∈ M for every t ≥ .Given a function ϕ ∈ Ξ( R n ) , the Musielak-Orlicz space is defined as the set L ϕ ( · ) ( R n ) := (cid:26) f ∈ M : lim λ → ρ ϕ ( · ) ( λf ) = 0 (cid:27) equipped with the (Luxemburg) norm k f k L ϕ ( · ) ( R n ) := inf { λ > ρ ϕ ( · ) ( x/λ ) ≤ } , where ρ ϕ ( · ) ( f ) = ˆ R n ϕ ( x, | f ( x ) | ) dx. Then L ϕ ( · ) ( R n ) is a Banach space. Many of the classical function spaces can also be viewed asMusielak-Orlicz spaces. • If ϕ ( x, t ) = t p with < p < ∞ , we get the classical Lebesgue space L p ( R n ) . • If ϕ ( x, t ) = ∞ (1 , ∞ ) ( t ) , then we get the Lebesgue space L ∞ ( R n ) . • If ϕ ( x, t ) = t p w ( x ) with < p < ∞ and a weight w , then we get the weighted Lebesguespace L p ( w ) . • If ϕ ( x, t ) = t p ( x ) , then we get the variable Lebesgue space L p ( · ) ( R n ) as in [12, Exam-ple 2.10] . • If ϕ ( x, t ) = ψ ( t ) , then we get the Orlicz spaces L ψ ( R n ) defined in (2.14) . • If ϕ ( x, t ) = t p (1 + log + t ) α with < p < ∞ and α ∈ R , then we get the Zygmund space L p (log L ) α see [8] . If we write X = L p (log L ) α , then p X = q X = p and X r = L pr (log L ) α forany < r < ∞ . Note that X r is a Banach space provided r is large enough (say, r > /p ). • If ϕ ( x, t ) = t p ( x ) (1+log + t ) q ( x ) , we get the generalization of the Zygmund spaces L p ( · ) (log L ) q ( · ) see [18] .Let us see when L ϕ ( · ) ( R n ) is a BFS. A function ϕ ∈ Ξ( R n ) is called proper if the set of simplefunctions L ( R n ) satisfies L ( R n ) ⊂ L ϕ ( · ) ( R n ) ∩ L ϕ ( · ) ( R n ) ′ . Diening et al. in [25, Section 2] proved that L ϕ ( · ) ( R n ) is a BFS if and only if ϕ is proper . Additionally, if ϕ ∈ Ξ( R n ) is proper, then L ϕ ( · ) ( R n ) ′ = L ϕ ∗ ( · ) ( R n ) , L ϕ ∗ ( · ) ( R n ) ′ = L ϕ ( · ) ( R n ) and L ϕ ( · ) ( R n ) ′′ = L ϕ ( · ) ( R n ) . For the boundedness of the maximal operator M , H¨ast¨o [35] showed that M : L ϕ ( · ) ( R n ) → L ϕ ( · ) ( R n ) , for all ϕ ∈ Ξ( R n ) with some extra assumptions. Extrapolation on Banach function spaces
This section is devoted to establishing the two-weight extrapolation theorems on Banachfunction spaces. Let us first recall the A p and A ∞ extrapolation in the one-weight case fromTheorem 3.9 and Corollary 3.15 in [21]. Theorem A.
If for some p ∈ [1 , ∞ ) and for every w ∈ A p , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F , then for every p ∈ (1 , ∞ ) and for every w ∈ A p , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F . Theorem B.
If for some p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F , then for every p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F . Given a weight v , we denote by M ′ v the dual of the Hardy-Littlewood maximal operator M . That is, M ′ v f ( x ) = M ( f v )( x ) /v ( x ) if v ( x ) = 0, M ′ v f ( x ) = 0 otherwise. Similarly, one candefine M ′ Φ ,v as the dual of the Orlicz maximal operator M Φ .3.1. From one-weight to two-weight.
We begin with two results about the so-called A p extrapolation. Theorem 3.1.
Suppose that X is a RIBFS over ( R n , dx ) with q X < ∞ . If for some p ∈ [1 , ∞ ) and for every w ∈ A p , k f k L ( w ) ≤ Ψ (cid:0) [ w ] A p (cid:1) k g k L ( w ) , ( f, g ) ∈ F , (3.1) where Ψ : [1 , ∞ ) → [1 , ∞ ) is an increasing function, then for every weight u such that u − p ′ ∈ A and for every v ∈ A ∩ L ( R n ) , k f u k X v ≤ (cid:16) k M v k X ′ v → X ′ v [ u − p ′ ] p − A [ v ] A (cid:17) k gu k X v , ( f, g ) ∈ F . (3.2) Here, u is understood as the constant in the case p = 1 .Proof. Fix u − p ′ ∈ A and v ∈ A ∩ L ( R n ). We claim that for every pair ( f, g ) ∈F with k f u k X v < ∞ and k gu k X v < ∞ , there exists w = w ( f, g ) ∈ A p with [ w ] A p ≤ k M v k X ′ v → X ′ v [ u − p ′ ] p − A [ v ] A such that k f u k X v ≤ k f k L ( w ) and k g k L ( w ) ≤ k gu k X v . (3.3)Assuming that our claim holds momentarily, let us see how (3.2) follows from (3.3). Given( f, g ) ∈ F , we may assume that k gu k X v < ∞ , otherwise, there is nothing to prove. We wouldlike to observe that f < ∞ a.e.. Otherwise, there exists a measurable set E ⊂ R n with | E | > f = ∞ on E . In view of (3.1), this gives that k g k L ( w ) = ∞ for every w ∈ A p . (3.4)On the other hand, applying (3.3) to f = g , we find a weight w = w ( g ) ∈ A p such that k g k L ( w ) ≤ k gu k X v < ∞ . This contradicts (3.4). WO-WEIGHT EXTRAPOLATION 15
For each N ≥
1, we define f N := f { B (0 ,N ): f ( x ) ≤ N,u ( x ) ≤ N } . The fact v ∈ L ( R n ) implies v ( B (0 , N )) < ∞ . Thanks to the property (6) in Definition 2.8, this gives that k f N u k X v ≤ N k B (0 ,N ) k X v < ∞ . Then applying (3.3), we get a weight w = w ( f N , g ) ∈ A p with [ w ] A p ≤ k M v k X ′ v → X ′ v [ u − p ′ ] p − A [ v ] A such that k f N u k X v ≤ k f N k L ( w ) and k g k L ( w ) ≤ k gu k X v . (3.5)Together with (3.1) and (3.5), it yields k f N u k X v ≤ k f N k L ( w ) ≤ k f k L ( w ) ≤ w ] A p ) k g k L ( w ) ≤ (cid:0) k M v k X ′ v → X ′ v [ u − p ′ ] p − A [ v ] A (cid:1) k gu k X v . (3.6)Recall that f < ∞ a.e.. Consequently, f N ր f as N → ∞ . Therefore, (3.6) and property (5)in Definition 2.8 immediately give (3.2) as desired.It remains to prove (3.3). Note that p X ′ = ( q X ) ′ >
1. Then Lemma 2.11 (d) gives that M v is bounded on X ′ v . Let h ∈ X ′ v be a nonnegative function such that h is non-zero on a set ofpositive measure. For the function h , we define the Rubio de Francia iteration algorithm as R v h := ∞ X j =0 M jv h j k M v k j X ′ v → X ′ v . (3.7)Then the following properties are fulfilled: h ≤ R v h, kR v h k X ′ v ≤ k h k X ′ v and [ R v h ] A ( v ) ≤ k M v k X ′ v → X ′ v . (3.8)Indeed, the first two inequalities are straightforward. Since h is nonnegative and non-zero ona set of positive measure, for every x ∈ R n there exists a cube Q x containing x such that ´ Q x hv dy >
0. By the fact that v is a weight with v ∈ L ( R n ), one has R v h ( x ) & M v h ( x ) = sup Q ∋ x v ( Q ) ˆ Q hv dy ≥ v ( Q x ) ˆ Q x hv dy > . (3.9)Let r >
0. Then v ( B (0 , r )) < ∞ , which together with Item (6) in Definition 2.8 implies k B (0 ,r ) k X v < ∞ . Hence, ˆ B (0 ,r ) R v h v dx ≤ k B (0 ,r ) k X v kR v h k X ′ v ≤ k B (0 ,r ) k X v k h k X ′ v < ∞ . Thus, R v h v < ∞ a.e. in B (0 , r ). By the arbitrariness of r , 0 < v < ∞ a.e. and (3.9), this inturn gives that 0 < R v h < ∞ a.e., that is, R v h is a weight. Moreover, the last one in (3.8)follows at once from the definition of R v .To proceed, fix ( f, g ) ∈ F with k f u k X v < ∞ . By (2.31) and Remark 2.9, there exists anonnegative function h ∈ X ′ v with k h k X ′ v ≤ h is non-zero on a set of positivemeasure and k f u k X v ≤ ˆ R n f ( x ) u ( x ) h ( x ) v ( x ) dx. (3.10)Pick w := u ( R v h ) v = ( u − p ′ ) − p ( R v h ) v . Since u − p ′ ∈ A , v ∈ A and R v h ∈ A ( v ) , by(2.6), (3.8) and Lemma 2.1 (5) we have[ w ] A p ≤ [ u − p ′ ] p − A [( R v h ) v ] A ≤ [ u − p ′ ] p − A [ R v h ] A ( v ) [ v ] A ≤ k M v k X ′ v → X ′ v [ u − p ′ ] p − A [ v ] A . In addition, the first one in (3.8) and (3.10) imply k f u k X v ≤ ˆ R n f ( x ) u ( x ) R v h ( x ) v ( x ) dx = 2 k f k L ( w ) . On the other hand, combining H¨older’s inequality (2.30) and (3.8), we obtain k g k L ( w ) = ˆ R n gu R v h v dx ≤ k gu k X v kR v h k X ′ v ≤ k gu k X v k h k X ′ v ≤ k gu k X v . This shows (3.3) and completes the proof. (cid:3)
Corollary 3.2.
Suppose that X be a RIBFS over ( R n , dx ) such that q X < ∞ . If for some p ∈ (1 , ∞ ) and for every w ∈ A p , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F . (3.11) then for every p ∈ (1 , ∞ ) , for every weight u such that u − p ′ ∈ A and for every weight v ∈ A ∩ L ( R n ) , k f p u k X v ≤ C k g p u k X v , ( f, g ) ∈ F . (3.12) Proof.
Fix p ∈ (1 , ∞ ). We first observe that the hypothesis (3.11) and Theorem A give thatfor every w ∈ A p , k f p k L ( w ) = k f k pL p ( w ) ≤ C k g k pL p ( w ) = k g p k L ( w ) , ( f, g ) ∈ F . This says that the hypothesis (3.1) is satisfied for the exponent p = p and the pair ( f p , g p ).Accordingly, (3.2) immediately implies (3.12) as desired. (cid:3) We now present an A extrapolation result. Theorem 3.3.
Suppose that u and v are weights on R n such that v ∈ A ∩ L ( R n ) , and X v is a BFS over ( R n , v dx ) . If for some p ∈ (0 , ∞ ) and for every w ∈ A , k f k L p ( w ) ≤ Ψ([ w ] A ) k g k L p ( w ) , ( f, g ) ∈ F , (3.13) where Ψ : [1 , ∞ ) → [1 , ∞ ) is a non-decreasing function, then k f u k X p v ≤ /p Ψ(2 K [ v ] A ) k gu k X p v , ( f, g ) ∈ F , (3.14) provided that k ( M v f ) u − p k X ′ v ≤ K k f u − p k X ′ v , ∀ f ∈ M . (3.15) Proof.
As we did in the proof of Theorem 3.1, it only needs to show that if (3.15) holds, then forevery pair ( f, g ) with k f u k X p v < ∞ and k gu k X p v < ∞ , there exists a weight w = w ( f, g ) ∈ A such that [ w ] A ≤ K [ v ] A , k f u k X p v ≤ /p k f k L p ( w ) and k g k L p ( w ) ≤ /p k gu k X p v . (3.16)To this end, fix a pair of functions ( f, g ) ∈ F with k f u k X p v < ∞ and k gu k X p v < ∞ . By(2.31), there exists a non-negative function h ∈ X ′ v with k h k X ′ v ≤ h is non-zero ona set of positive measure, and k f u k p X p v = k f p u p k X v ≤ ˆ R n f p u p hv dx. (3.17)For this function h , we define R h := ∞ X j =0 M jv h j K j and H := R ( hu p ) u − p . WO-WEIGHT EXTRAPOLATION 17
Then using (3.15), one can verify that h ≤ H, k H k X ′ v ≤ k h k X ′ v and [ R h ] A ( v ) ≤ K . (3.18)If we set w := u p Hv , then v ∈ A and Lemma 2.1 (6) imply [ w ] A = [ R ( hu p ) v ] A ≤ [ R ( hu p )] A ( v ) [ v ] A ≤ K [ v ] A . It follows from (3.17) and (3.18) that k f u k p X p v ≤ ˆ R n f p u p Hv dx = 2 k f k p L p ( w ) . Also, invoking (2.30) and (3.18), we conclude that k g k p L p ( w ) = ˆ R n g p u p Hv dx ≤ k g p u p k X v k H k X ′ v ≤ k gu k p X p v This shows (3.16). (cid:3)
Next, we turn our attention to A ∞ extrapolation. Theorem 3.4.
Suppose that u and v are weights on R n such that v ∈ A ∞ ∩ L ( R n ) , and X v is a BFS over ( R n , v dx ) . If for some p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F , (3.19) then for every p ∈ (0 , ∞ ) , k f p u k X v ≤ C k g p u k X v , ( f, g ) ∈ F , (3.20) provided that k ( M v f ) u − k X ′ v ≤ K k f u − k X ′ v , ∀ f ∈ M . (3.21) Moreover, under the hypothesis (3.21) , (3.19) implies that for every p, q ∈ (0 , ∞ ) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) X j f qj (cid:17) pq u (cid:13)(cid:13)(cid:13)(cid:13) X v ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) X j g qj (cid:17) pq (cid:13)(cid:13)(cid:13)(cid:13) X v , { ( f j , g j ) } j ⊂ F . (3.22) Proof.
By Theorem B , we have that for every p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F . (3.23)This can be rewritten as k f p k L ( w ) ≤ C k g p k L ( w ) , ( f, g ) ∈ F . (3.24)As before, it is enough to prove that if (3.21) holds, then for every pair ( f, g ) ∈ F with k f u k X v < ∞ and k gu k X v < ∞ , there exists a weight w = w ( f, g ) ∈ A ∞ such that [ w ] A ∞ ≤ K [ v ] A ∞ , k f u k X v ≤ k f k L ( w ) and k g k L ( w ) ≤ k gu k X v . (3.25)We here point out that (3.20) will follow from (3.24) and (3.25).Let us turn to the proof of (3.25) and it is similar to that of (3.16). In the current setting,the Rubio de Francia iteration algorithm is replaced by R h := ∞ X j =0 M jv h j K j and H := R ( hu ) u − . Then it follows from (3.21) that h ≤ H, k H k X ′ v ≤ k h k X ′ v and [ R h ] A ( v ) ≤ K . (3.26) Pick w := uHv . Then by the last estimate in (3.26), v ∈ A ∞ and Lemma 2.1 (6), we obtainthat w = R ( hu ) v ∈ A ∞ with [ w ] A ∞ ≤ [ R ( hu )] A ( v ) [ v ] A ∞ ≤ K [ v ] A ∞ . The rest of argumentis almost the same as in Theorem 3.3.Additionally, (3.22) is a consequence of (3.20) and (3.23). (cid:3) Theorem 3.5.
Suppose that X is a RIBFS over ( R n , dx ) such that q X < ∞ . If for some p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F , (3.27) then for every u ∈ RH ∞ and for every v ∈ A ∞ ∩ L ( R n ) , k f u k X v ≤ C k gu k X v , ( f, g ) ∈ F . (3.28) Proof.
By Theorem B , the hypothesis (3.27) implies that for every p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F . (3.29)It suffices to show that for every r ∈ (1 , ∞ ), for every pair ( f, g ) ∈ F with k f u k X v < ∞ and k gu k X v < ∞ , there exists a weight w = w ( f, g ) ∈ A ∞ such that k f u k X v ≤ r k f k L r ( w ) and k g k L r ( w ) ≤ r k gu k X v . (3.30)Let u ∈ RH ∞ and v ∈ A ∞ . Denote Y = X r . Then p Y ′ = ( q Y ) ′ = ( rq X ) ′ > q X < ∞ . Then, it follows from Lemma 2.11 (d) that M v is bounded on Y ′ v . For any non-negative function h ∈ Y ′ v such that h is non-zero on a set of positive measure, we define theRubio de Francia iteration algorithm as: R v h := ∞ X j =0 M kv h k k M v k k Y ′ v → Y ′ v . (3.31)Then it is easy to verify that h ≤ R v h, kR v h k Y ′ v ≤ k h k Y ′ v and [ R v h ] A ( v ) ≤ k M v k Y ′ v . (3.32)From (2.31), there exists a nonnegative function h ∈ Y ′ v with k h k Y ′ v ≤ h isnon-zero on a set of positive measure, and k f u k r X v = (cid:13)(cid:13) ( f u ) r (cid:13)(cid:13) Y v ≤ ˆ R n f r u r hv dx. (3.33)Since v ∈ A ∞ and R v h ∈ A ( v ), Lemma 2.1 (6) implies ( R v h ) v ∈ A ∞ . Also, by Lemma 2.1(3), one has u r ∈ RH ∞ . Together with Lemma 2.1 (2), these facts give that w := u r ( R v h ) v ∈ A ∞ . (3.34)Moreover, by (3.32) and (3.33), k f u k X v ≤ r k f k L r ( w ) , and k g k r L r ( w ) = ˆ R n g r u r R v h v dx ≤ k ( gu ) r k Y v kR v h k Y ′ v ≤ k gu k r X v . The proof is complete. (cid:3)
WO-WEIGHT EXTRAPOLATION 19
Theorem 3.6.
Suppose that X is a RIBFS over ( R n , dx ) such that q X < ∞ . If for some p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F , (3.35) then for every u ∈ A and for every v ∈ A ∞ ∩ L ( R n ) , or for every u ∈ A ∞ and for every v ∈ A ∩ L ( R n ) , (cid:13)(cid:13)(cid:13) fu (cid:13)(cid:13)(cid:13) X v ≤ C (cid:13)(cid:13)(cid:13) gu (cid:13)(cid:13)(cid:13) X v , ( f, g ) ∈ F . (3.36) Proof.
Let u ∈ A and v ∈ A ∞ ∩ L ( R n ). Then Lemma 2.1 (1) implies u − ∈ RH ∞ .Thus, (3.36) follows from Theorem 3.5. In what follows, we assume that u ∈ A ∞ and v ∈ A ∩ L ( R n ).By Theorem B , the hypothesis (3.27) implies that for every p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F . (3.37)It suffices to show that for every pair ( f, g ) ∈ F with k f u k X v < ∞ and k gu k X v < ∞ , thereexist some r ∈ (1 , ∞ ) and a weight w = w ( f, g ) ∈ A ∞ such that k f u k X v ≤ r k f k L r ( w ) and k g k L r ( w ) ≤ r k gu k X v . (3.38)Denote Y = X r for some r ∈ (1 , ∞ ) chosen later. For any non-negative function h ∈ Y ′ v such that h is non-zero on a set of positive measure, we define the Rubio de Francia iterationalgorithm as: R v h := ∞ X j =0 M kv h k k M v k k Y ′ v → Y ′ v . (3.39)Then we have h ≤ R v h, kR v h k Y ′ v ≤ k h k Y ′ v and [ R v h ] A ( v ) ≤ k M v k Y ′ v (3.40)By (2.31), there exists a nonnegative function h ∈ Y ′ v with k h k Y ′ v ≤ h is non-zeroon a set of positive measure, and (cid:13)(cid:13)(cid:13) fu (cid:13)(cid:13)(cid:13) r X v = (cid:13)(cid:13)(cid:13)(cid:16) fu (cid:17) r (cid:13)(cid:13)(cid:13) Y v ≤ ˆ R n f r u − r hv dx. Recall that u ∈ A ∞ and v ∈ A . It follows from [34, Theorem 1.3] that there exists adimensional constant c n > u ] A p ≤ e e cn [ u ] A ∞ , ∀ p > e c n [ u ] A ∞ . (3.41)Pick p = 1 + e c n [ u ] A ∞ such that u ∈ A p . Then by the A p factorization theorem, one has u = u u − p for some u , u ∈ A . We claim that there exists C n > w := u − r R v h · v = (cid:20) ( R v h · v ) u p − r (cid:21) u − r ∈ A ∞ , ∀ r > C n [ v ] A e C n [ u ] A ∞ . (3.42)Indeed, a straightforward calculation gives that σ := R v h · v ∈ A with[ σ ] A ≤ [ v ] A [ R v h ] A ( v ) ≤ v ] A k M v k Y ′ v → Y ′ v ≤ C [ v ] A sup Q ⊂ R n v (3 Q ) v ( Q ) ≤ C · n [ v ] A , (3.43) where we used Lemma 2.1 (5), (3.40), (2.40) and the doubling property: for any ω ∈ A s ,1 ≤ s < ∞ , ω ( λQ ) ≤ λ ns [ ω ] A s ω ( Q ) , ∀ λ > ∀ Q ⊂ R n . It follows from (5.3) below that for every σ ∈ A and for every cube Q ⊂ R n , (cid:18) Q σ ( x ) r σ dx (cid:19) /r σ ≤ Q σ ( x ) dx, where r σ := 1 + 12 n +1 [ σ ] A . Then by (3.43), we have r ′ σ ( p −
1) = (1 + 2 n +1 [ σ ] A ) e c n [ u ] A ∞ ≤ e c n [ v ] A e e c n [ u ] A ∞ . Let r > C n [ v ] A e C n [ u ] A ∞ with C n ≥ e c n . Then r ′ σ ( p − ≤ r and Q σu p − r dx ≤ (cid:18) Q σ r σ dx (cid:19) rσ (cid:18) Q u r ′ σ ( p − r dx (cid:19) r ′ σ ≤ (cid:18) Q σ dx (cid:19) (cid:18) Q u dx (cid:19) p − r ≤ (cid:16) [ σ ] A inf Q σ (cid:17)(cid:16) [ u ] A inf Q u (cid:17) p − r ≤ σ ] A [ u ] p − r A inf Q (cid:16) σu p − r (cid:17) . That is, σu p − r ∈ A . On the other hand, u ∈ A implies that u r ∈ A with [ u r ] A ≤ [ u ] r A . Inview of Lemma 2.1 (1), one has u − r ∈ RH ∞ . Therefore, these and Lemma 2.1 (4) immediatelyyield (3.42).The remaining argument is almost the same in the proof of Theorem 3.5. We omit thedetails. (cid:3) Theorem 3.7.
Suppose that u and v are weights such that v ∈ L ( R n ) , and X v is a BFSover ( R n , v dx ) . Assume that there exit r > and s > such that u s v − sr ′ ∈ A and k M ′ v /r f k X ′ v ≤ K k f k X ′ v , ∀ f ∈ M . (3.44) If for some p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F , then (cid:13)(cid:13)(cid:13) fu (cid:13)(cid:13)(cid:13) X v ≤ C (cid:13)(cid:13)(cid:13) gu (cid:13)(cid:13)(cid:13) X v , ( f, g ) ∈ F . Proof.
Let h be a non-negative function such that h ∈ X ′ v and h is non-zero on a set of positivemeasure. In view of (3.44), we define the Rubio de Francia algortihm as: R h := ∞ X j =0 ( M ′ v /r ) k h k k M ′ v /r k k X ′ v → X ′ v . Then we see that R h · v r ∈ A . Combining this with u s v − sr ′ ∈ A and (2.6), we obtain w := u − ( R h ) v = ( u s v − sr ′ ) − ( s +1) ( R h · v r ) ∈ A s +1 ⊂ A ∞ . (3.45)The remaining argument is the same as we did before. The only difference is that we do notuse the rescaling argument in this case. (cid:3) WO-WEIGHT EXTRAPOLATION 21
Theorem 3.8.
Suppose that u and v are weights on R n such that v ∈ A ∞ and uv ∈ L ( R n ) ,and X uv is a BFS over ( R n , uv dx ) . Assume that there exist q > and K > such that k M ′ u f k ( X q ′ uv ) ′ ≤ K k f k ( X q ′ uv ) ′ , ∀ q ≥ q . (3.46) If for some p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F . (3.47) then (cid:13)(cid:13)(cid:13) fv (cid:13)(cid:13)(cid:13) X uv ≤ C (cid:13)(cid:13)(cid:13) gv (cid:13)(cid:13)(cid:13) X uv , ( f, g ) ∈ F , (3.48) Remark 3.9.
Under the same hypotheses, one can also obtain that there exists a small constant ǫ > depending on only q and [ v ] A ∞ such that (cid:13)(cid:13)(cid:13) fv (cid:13)(cid:13)(cid:13) X ǫuv ≤ C (cid:13)(cid:13)(cid:13) gv (cid:13)(cid:13)(cid:13) X ǫuv , ( f, g ) ∈ F . (3.49) In fact, in the proof below, it suffices to pick r > and ǫ > such that < r ′ < min { ε , q } and r (1 + ǫ ) < q ′ . On the other hand, if the assumption v ∈ A ∞ is replaced by v ∈ RH ∞ , thenthe conclusion (3.48) still holds provided a weaker condition than (3.46) : k M ′ u f k ( X ruv ) ′ ≤ K k f k ( X ruv ) ′ for some r > . Proof.
For every U ∈ A with [ U ] ≤ K , by Lemma 2.1 (7), there exists ε := ε ( K ) ∈ (0 , U V ε ∈ A for every V ∈ A and ε ∈ (0 , ε ) . (3.50)Pick r > < r ′ < min { ε , q } . Denote Y uv = X ruv . Since r ′ > q , the hypothesis(3.46) implies that M ′ u is bounded on Y ′ uv with bound K .For any non-negative function h ∈ Y ′ uv with k h k Y ′ uv ≤ h is non-zero on a set ofpositive measure, we define the Rubio de Francia iteration algorithm as: R h := ∞ X j =0 ( M ′ u ) j h j K j . Then one can check that h ≤ R h, kR h k Y ′ uv ≤ k h k Y ′ uv and [ R h · u ] A ≤ K . (3.51)On the other hand, since v ∈ A ∞ , it follows from Lemma 2.1 (4) that v = v v for some v ∈ A and v ∈ RH ∞ . Invoking (3.50) and (3.51), we get( R h · u ) v r ′ ∈ A . (3.52)Note that v r ′ ∈ RH ∞ by Lemma 2.1 (3). Accordingly, combining (3.52) with Lemma 2.1 (4),this gives that w := R h · uv r ′ = [( R h · u ) v r ′ ] v r ′ ∈ A ∞ . (3.53)The rest of the proof is similar to the argument in the proof of Theorem 3.6. (cid:3) Theorem 3.10.
Suppose that X is a RIBFS over ( R n , dx ) with q X < ∞ . If for some p ∈ (0 , ∞ ) and every w ∈ A ∞ , k f k L p ( w ) ≤ C k M g k L p ( w ) , ( f, g ) ∈ F , (3.54) then for every weight w , (cid:13)(cid:13)(cid:13)(cid:13) fM w (cid:13)(cid:13)(cid:13)(cid:13) X ( Mw ) ≤ C (cid:13)(cid:13)(cid:13) gM w (cid:13)(cid:13)(cid:13) X ( Mw ) , ( f, g ) ∈ F . (3.55) Proof.
We first note that the hypothesis (3.54) and Theorem B give that for every p ∈ (0 , ∞ )and every w ∈ A ∞ , k f k L p ( w ) ≤ C k M g k L p ( w ) , ( f, g ) ∈ F . (3.56)It suffices to show that for every (or for some) s > f, g ) ∈ F with (cid:13)(cid:13)(cid:13) fM w (cid:13)(cid:13)(cid:13) X ( Mw ) < ∞ and (cid:13)(cid:13) gMw (cid:13)(cid:13) X ( Mw ) < ∞ , there exists a weight w = w ( f, g ) ∈ A ∞ such that (cid:13)(cid:13)(cid:13)(cid:13) fM w (cid:13)(cid:13)(cid:13)(cid:13) X ( Mw ) . k f k L s ( w ) and k M g k L s ( w ) . (cid:13)(cid:13)(cid:13) gM w (cid:13)(cid:13)(cid:13) X ( Mw ) . (3.57)Fix s > Y = X s . For each r >
1, (
M w ) − r (1 − s ) = [( M w ) − s ] − r ( M w ) s ∈ A r . Ifwe denote v := ( M w ) − s , then k M ′ v f k L r ( Mw ) = (cid:13)(cid:13) M ( f v ) (cid:13)(cid:13) L r (( Mw ) − r (1 − s ) ) . (cid:13)(cid:13) f v (cid:13)(cid:13) L r (( Mw ) − r (1 − s ) ) = k f k L r ( Mw ) . (3.58)Note that p Y ′ = ( q Y ) ′ = ( sq X ) ′ > q Y ′ = ( p Y ) ′ = ( sp X ) ′ < ∞ . Taking r , r ∈ (1 , ∞ ) such that 1 < r < p Y ′ ≤ q Y ′ < r < ∞ , we obtain from (3.58) that M ′ v is bounded on L r ( M w ) and L r ( M w ). Hence, Lemma 2.10 gives that M ′ v : Y ′ ( M w ) → Y ′ ( M w ) boundedly . Let h ∈ Y ′ ( M w ) be a nonnegative function such that h is non-zero on a set of positive measure.Now we define the Rubio de Francia iteration algorithm as R h := ∞ X j =0 ( M ′ v ) k h k k M ′ v k Y ′ ( Mw ) . As before, it is easy to verify that h ≤ R h, kR h k Y ′ ( Mw ) ≤ k h k Y ′ ( Mw ) and [ R h · v ] A ≤ k M ′ v k Y ′ ( Mw ) → Y ′ ( Mw ) . (3.59)To proceed, we present several facts for the maximal operator M . First, we have M f ( x ) = sup Q ∋ x Q | f ( y ) | dy ≤ sup Q ∋ x n Q ( x,ℓ ( Q )) | f ( y ) | dy = sup Q ∋ x n Q ( x,ℓ ( Q )) | f | dw Q ( x,ℓ ( Q )) w dy . M cw ( f w − )( x ) M w ( x ) , (3.60)where Q ( x, ℓ ( Q )) denotes the cube centered at x with side-length 2 ℓ ( Q ). Gathering (3.60) andLemma 2.11 (c), we obtain (cid:13)(cid:13)(cid:13)(cid:13) M fM w (cid:13)(cid:13)(cid:13)(cid:13) X ( w ) . k M cw ( f w − ) k X ( w ) . k f w − k X ( w ) . (3.61)On the other hand, as we see in [50, eq.(2.2)], (cid:18) M fM f (cid:19) ≤ C n M fM f . (3.62) WO-WEIGHT EXTRAPOLATION 23
By (2.31), there exists a nonnegative function h ∈ Y ′ ( M w ) with k h k Y ′ ( Mw ) ≤ h is non-zero on a set of positive measure, and (cid:13)(cid:13)(cid:13)(cid:13) fM w (cid:13)(cid:13)(cid:13)(cid:13) s X ( Mw ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) fM w (cid:19) s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y ( Mw ) ≤ ˆ R n f s h M w ( M w ) s dx. (3.63)Since ( M w ) s ∈ A by (2.7), it follows from (2.6) and (3.59) that w := ( R h · v )( M w ) − s = ( R h · v )[( M w ) s ] − ∈ A . (3.64)Thanks to (3.63) and (3.59), one has (cid:13)(cid:13)(cid:13)(cid:13) fM w (cid:13)(cid:13)(cid:13)(cid:13) s X ( Mw ) . ˆ R n f s ( R h ) M w ( M w ) s dx ≤ ˆ R n f s ( R h )( M w ) − s ( M w ) s dx = k f k s L s ( w ) . Sequently, applying (3.62) and the H¨older’s inequality (2.30) for Y ( M w ), we deduce that k M g k s L s ( w ) = ˆ R n ( M g ) s ( R h )( M w ) − s ( M w ) s dx . ˆ R n (cid:18) M gM w (cid:19) s ( R h ) M w dx ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) M gM w (cid:19) s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y ( Mw ) kR h k Y ′ ( Mw ) . (cid:13)(cid:13)(cid:13)(cid:13) M gM w (cid:13)(cid:13)(cid:13)(cid:13) s X ( Mw ) k h k Y ′ ( Mw ) . (cid:13)(cid:13)(cid:13) gM w (cid:13)(cid:13)(cid:13) s X ( Mw ) , where (3.61) was used in the last step. This proves (3.57). (cid:3) From two-weight to two-weight.Theorem 3.11.
Suppose that X is a RIBFS over ( R n , dx ) with q X < ∞ such that X p is aBFS. If for some p ∈ (0 , ∞ ) and for every weight w , k f k L p ( w ) ≤ C k g k L p ( Mw ) , ( f, g ) ∈ F , (3.65) then for every weight w , k f k X w ≤ C k g ( M w/w ) p k X w , ( f, g ) ∈ F . (3.66) Moreover, if X q is a BFS for some q ∈ (0 , ∞ ) , then (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ∞ X j =1 | f j | q (cid:19) q (cid:13)(cid:13)(cid:13)(cid:13) X w ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ∞ X j =1 | g j | q (cid:19) q (cid:18) M ww (cid:19) q (cid:13)(cid:13)(cid:13)(cid:13) X w , { ( f j , g j ) } j ⊂ F . (3.67) Proof.
Let Y = X p . Then Y is a BFS and k f k p X w = k f p k Y w = sup ≤ h ∈ Y ′ w k h k Y ′ w ≤ ˆ R n f p hw dx. (3.68) Fix a nonnegative function h ∈ Y ′ w with k h k Y ′ w ≤ . Note that q X < ∞ implies that p Y ′ =( q Y ) ′ = ( q X /p ) ′ >
1. Invoking (3.65), (3.60) and Lemma 2.11 (c), we conclude that ˆ R n f p hw dx . ˆ R n g p M ( hw ) dx ≤ ˆ R n g p M w M cw h dx = ˆ R n g p ( M w/w ) M cw h w dx ≤ k g p ( M w/w ) k Y w k M cw h k Y ′ w . k g ( M w/w ) p k p X w k h k Y ′ w . (3.69)Therefore, (3.66) as desired follows from (3.68) and (3.69). The inequality (3.67) is a conse-quence of (3.66). Indeed, given q ∈ (0 , ∞ ), we define a new family F q by F q = (cid:26) ( F, G ) := (cid:18)(cid:16) ∞ X j =1 | f j | q (cid:17) q , (cid:16) ∞ X j =1 | g j | q (cid:17) q (cid:19) : { ( f j , g j ) } j ⊂ F (cid:27) . Then by (3.65), one has for every weight w and every pair ( F, G ) ∈ F q , k F k L q ( w ) = (cid:18) ∞ X j =0 k f j k qL q ( w ) (cid:19) q ≤ C (cid:18) ∞ X j =0 k g j k qL q ( Mw ) (cid:19) q = k G k L q ( Mw ) . (3.70)This shows (3.65) holds for the exponent p = q and the pair ( f, g ) = ( F, G ). Thus, (3.66)immediately implies (3.67). (cid:3)
Theorem 3.12.
Let u , v , w and w be weights on R n . Let Φ be a Young function or Φ( t ) = t .Suppose that X u and X v are respectively BFS over ( R n , u dx ) and ( R n , v dx ) such that Y u = X p u and Y v = X p v are BFS for some p ∈ (0 , ∞ ) . Assume that k ( M Φ f ) w − p v − k Y ′ v ≤ C k f w − p u − k Y ′ u , ∀ f ∈ M . (3.71) If for every weight w , k f k L p ( w ) ≤ C k g k L p ( M Φ w ) , ( f, g ) ∈ F . (3.72) then k f w k X u ≤ C k gw k X v , ( f, g ) ∈ F . (3.73) Proof.
Recall that Y u and Y v are Banach function spaces. Then by (2.31), k f w k p X u = k f p w p k Y u = sup ≤ h ∈ Y ′ u k h k Y ′ u ≤ ˆ R n f p w p hu dx. (3.74)Now fix a nonnegative function h ∈ Y ′ u such that k h k Y ′ u ≤
1. By the hypothesis (3.71), (3.72)and H¨older’s inequality (2.30), we have ˆ R n f p w p hu dx . ˆ R n g p M Φ ( w p hu ) dx = ˆ R n g p w p · M Φ ( w p hu ) w − p dx ≤ k g p w p k Y v k M Φ ( w p hu ) w − p v − k Y ′ v . k gw k p X v k h k Y ′ v . This and (3.74) immediately give (3.73). (cid:3)
WO-WEIGHT EXTRAPOLATION 25
Remark 3.13.
Theorem . extends [21, Theorem 8.2] to the weighted Banach functionspaces. Indeed, let A , B and Φ be Young functions such that A ∈ B p and A − ( t ) B − ( t ) . Φ − ( t ) for any t > . Let u = v , X u = X v = L p ( v ) , r = p/p > and ( w , w ) = ( U p , V p ) .With these notation in hand, we see that Y ′ u = Y ′ v = L r ′ ( v ) and (3.71) is equivalent to k M Φ f k L r ′ (( vV ) − r ′ ) . k f k L r ′ (( vU ) − r ′ ) . (3.75) Moreover, (3.73) is equivalent to k f k L p ( vU ) ≤ C k g k L p ( vV ) , ( f, g ) ∈ F . In view of Lemma . , the inequality (3.75) holds if sup Q k ( vU ) r k A,Q k ( vV ) − r k r ′ ,Q = sup Q k (( vV ) − r ′ ) r ′ k r ′ ,Q k (( vU ) − r ′ ) − r ′ k A,Q < ∞ . Theorem 3.14.
Let Φ be a Young function or Φ( t ) = t , A and B be Young functions suchthat A − ( t ) B − ( t ) . Φ − ( t ) . Suppose that u and v are weights on R n , and X v and X M A v areBFS over ( R n , v dx ) and over ( R n , M A v dx ) respectively. Assume that for for some p ∈ (0 , ∞ ) and for every weight w , k f k L p ( w ) ≤ C k g k L p ( M Φ w ) , ( f, g ) ∈ F . (3.76)(i) If Y v = X p v is a BFS such that k M ′ B,v f k Y ′ v ≤ C k f k Y ′ v , ∀ f ∈ M , (3.77) then k f u k X v ≤ C k gM A ( u p ) p k X v , ( f, g ) ∈ F . (3.78)(ii) If both Y v = X p v and Y M A v = X p M A v are BFS such that k M ′ B,u p f k Y ′ MAv ≤ C k f k Y ′ v , ∀ f ∈ M , (3.79) then k f u k X v ≤ C k gu k X MAv , ( f, g ) ∈ F . (3.80) Proof.
Note that Y v = X p v and Y M A v = X p M A v are BFS. Thanks to (2.31), we see that k f u k p X v = k f p u p k Y v = sup ≤ h ∈ Y ′ v k h k Y ′ v ≤ ˆ R n f p u p hv dx. (3.81)Let 0 ≤ h ∈ Y ′ v with k h k Y ′ v ≤
1. By our assumption on Φ and (2.16), one has M Φ ( f g )( x ) . M A f ( x ) M B g ( x ) , ∀ x ∈ R n . (3.82)Thus, applying the hypothesis (3.76), (3.82) and H¨older’s inequality (2.30), we obtain ˆ R n f p u p hv dx . ˆ R n g p M Φ ( u p hv ) dx . ˆ R n g p M A ( u p ) M B ( hv ) dx ≤ k g p M A ( u p ) k Y v k M B ( hv ) v − k Y ′ v . k gM A ( u p ) p k p X v k h k Y ′ v , (3.83) where (3.77) was used in the last step. On the other hand, by (3.79), ˆ R n f p u p hv dx . ˆ R n g p M Φ ( u p hv ) dx . ˆ R n g p M B ( u p h ) M A v dx ≤ k g p u p k Y MAv k M B ( u p h ) u − p k Y ′ MAv . k gu k p X MAv k h k Y ′ v . (3.84)Consequently, (3.78) and (3.79) follow at once from (3.81), (3.83) and (3.84). (cid:3) The classical Fefferman-Stein inequality in [28] asserts that for each p ∈ (1 , ∞ ), k M f k L p ( w ) ≤ C p k f k L p ( Mw ) for every weight w. (3.85)In [62, Theorem 1.7], P´erez proved that for any 1 < p < ∞ , Φ ∈ B p if and only if k M Φ f k L p ( w ) ≤ C k f k L p ( Mw ) for every weight w. (3.86)Another result in [62, Theorem 6.1] states that r < p if and only if k M r,s f k L p ( w ) ≤ C k f k L p ( Mw ) for every weight w, (3.87)whenever 1 < p, r < ∞ and 1 ≤ s < ∞ , where M r,s is the maximal operator associated toLorentz space L r,s ( R n ) defined by M r,s f ( x ) := sup Q ∋ x | Q | − r k f Q k L r,s ( R n ) . Moreover, note that for any Young function A , M ( f g )( x ) ≤ M A f ( x ) M ¯ A g ( x ) , ∀ x ∈ R n . (3.88)As a consequence, by (3.85)–(3.88) and the approach used in (3.70), Theorem 3.14 gives thefollowing estimates. Theorem 3.15.
Let < q < ∞ and v be a weight on R n . Suppose that X v is a BFS over ( R n , v dx ) such that X q v is also a BFS. Let A be a Young function. If M ′ ¯ A,v is bounded on ( X q v ) ′ , then for every weight u , (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ∞ X j =1 | M f j | q (cid:17) q u (cid:13)(cid:13)(cid:13)(cid:13) X v ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ∞ X j =1 | f j | q (cid:17) q M A ( u q ) q (cid:13)(cid:13)(cid:13)(cid:13) X v , (3.89) where the maximal operator M is one of the following: (1) M = M ; (2) M = M Φ with Φ ∈ B p for some p ∈ (1 , ∞ );(3) M = M r,s with < r < ∞ and ≤ s < ∞ . Now let us see how Theorem 3.15 recovers the sharp vector-valued inequality obtained in[23, Theorem 1.4] and [63, Theorem 1.1]. Denote r = p/q, ε = [ r ] + 1 − r, A ( t ) = A ( t /r ) and A ( t ) = t r log( e + t ) r − ε . Then, one has A ( t ) ≃ t log( e + t ) [ r ] and ¯ A ( t ) ≃ t r ′ log( e + t ) − − ( r ′ − ε ∈ B r ′ . (3.90)Note that for Φ k ( t ) = t log( e + t ) k , M k +1 f ( x ) ≃ M Φ k f ( x ) , x ∈ R n and k ∈ N + . (3.91) WO-WEIGHT EXTRAPOLATION 27
Let w be an arbitrary weight. Thus, the former in (3.90) and (3.91) imply M A ( w /r )( x ) r = M A w ( x ) ≃ M [ r ]+1 w ( x ) . Also, it follows from (3.90) and Lemma 2.4 that M ¯ A is bounded on L r ′ ( R n ). Accordingly, theinequality (3.89) applied to u = w p , v ≡ X v = L p ( R n ) gives the following result. Corollary 3.16.
Let < q < p < ∞ . There exists a constant C = C ( n, p, q ) such that forevery weight w , (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ∞ X j =1 | M f j | q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p ( w ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ∞ X j =1 | f j | q (cid:17) q (cid:13)(cid:13)(cid:13)(cid:13) L p ( M [ p/q ]+1 w ) . (3.92) Moreover, the inequality (3.92) is sharp in the sense that the exponent [ p/q ] + 1 cannot bereplaced by [ p/q ] . Extrapolation on modular spaces
We turn now our attention to the extrapolation on modular spaces. In this context, for aweight w and a Young function Φ, we define the modular ρ Φ w of f ∈ M by ρ Φ w ( f ) := ˆ R n Φ( | f ( x ) | ) w ( x ) dx. When w ≡
1, we denote ρ Φ instead of ρ Φ w . The collection of functions, M Φ w := { f ∈ M : ρ Φ w ( f ) < ∞} . is referred to as a modular space . Theorem 4.1.
Let Φ be a Young function with < i Φ ≤ I Φ < ∞ . If for some p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F , (4.1) then for every u ∈ RH ∞ and v ∈ A ∞ ∩ L ( R n ) , ˆ R n Φ( f u ) v dx ≤ C ˆ R n Φ( gu ) v dx, ( f, g ) ∈ F . (4.2) Moreover, the following vector-valued inequality holds for every q ∈ (0 , ∞ ) , ˆ R n Φ (cid:18)(cid:16) X j f qj (cid:17) q u (cid:19) v dx ≤ C ˆ R n Φ (cid:18)(cid:16) X j g qj (cid:17) q u (cid:19) v dx, { ( f j , g j ) } ⊂ F . (4.3) Proof.
We begin with a claim: for every pair ( f, g ) ∈ F with ρ Φ v ( f u ) < ∞ and ρ Φ v ( gu ) < ∞ ,there exists a weight w = w ( f, g ) ∈ A ∞ such that ρ Φ v ( f u ) ≤ k f k L ( w ) and k g k L ( w ) ≤ C (cid:0) ερ Φ v ( f u ) + ε − r ρ Φ v ( gu ) (cid:1) , (4.4)where ε ∈ (0 ,
1) is an arbitrary number, C > r > ε . Assumingthis momentarily, let us see how (4.2) follows from (4.4). Observe that by Theorem B , (4.1)implies that for every p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F . (4.5) Given ( f, g ) ∈ F , we may assume that ρ Φ v ( gu ) < ∞ . This in turn implies f < ∞ a.e..Otherwise, there exists a measurable set E ⊂ R n with | E | > f = ∞ on E . Bymeans of (4.5), it holds k g k L ( w ) = ∞ for every w ∈ A ∞ . (4.6)On the other hand, applying our claim to f = g and ε = 1 /
2, we get a weight w = w ( g ) ∈ A ∞ such that k g k L ( w ) ≤ C r +1 ρ Φ v ( gu ) < ∞ , which contradicts (4.6).For every N ≥
1, we define f N := f { x ∈ B (0 ,N ): f ( x ) ≤ N,u ( x ) ≤ N,v ( x ) ≤ N } . This gives that ρ Φ v ( f N u ) ≤ N Φ( N ) | B (0 , N ) | < ∞ . Then (4.4) applied to the pair ( f N , g ) give that w = w ( f N , g ) ∈ A ∞ ρ Φ v ( f N u ) ≤ k f N k L ( w ) and k g k L ( w ) ≤ C (cid:0) ερ Φ v ( f N u ) + ε − r ρ Φ v ( gu ) (cid:1) , (4.7)where ε ∈ (0 ,
1) is an arbitrary number, C > r > ε and N .Gathering (4.5) and (4.7), we conclude that ρ Φ v ( f N u ) ≤ k f N k L ( w ) ≤ k f k L ( w ) ≤ C k g k L ( w ) ≤ C C (cid:0) ερ Φ v ( f N u ) + ε − r ρ Φ v ( gu ) (cid:1) ≤ ρ Φ v ( f N u ) + C C ε − r ρ Φ v ( gu ) , (4.8)provided 0 < ε < C C . Therefore, (4.8) gives that ρ Φ v ( f N u ) ≤ C C ε − r ρ Φ v ( gu ) . (4.9)Recall that f < ∞ a.e., and hence f N ր f as N → ∞ . Consequently, (4.2) follows at oncefrom (4.9) and the monotone convergence theorem. The vector-valued inequality (4.3) can beshown as before.In order to show (4.4), we invoke a result contained in [24, Proposition 5.1]. That is, forevery φ ∈ ∆ and for every w ∈ A ∞ , ˆ R n φ ( M w f ( x )) w ( x ) dx ≤ K ˆ R n φ ( | f ( x ) | ) w ( x ) dx, (4.10)where K := K ( φ, w ) ≥ f . For a non-negative function h with ρ ¯Φ v ( h ) < ∞ such that h is non-zero on a set of positive measure, we define the Rubio de Francia iterationalgorithm as: R v h := ∞ X j =0 M jv h j K j , where K = K ( ¯Φ , v ) is defined in (4.10). The following properties are verified: h ≤ R v h, ρ ¯Φ v ( R v h ) ≤ ρ Φ v (2 h ) and [ R h ] A ( v ) ≤ K . (4.11)The proof of the first and the last estimates in (4.11) is the same as for the analogue propertiesin the proof of Theorem 3.1. Now write λ := K ≤ (see [12, Lemma 4.2]). From this andthe formula for the sum of a geometric series, we obtain that ∞ X k =0 (1 − λ ) λ k = 1 . WO-WEIGHT EXTRAPOLATION 29
In addition, by the convexity of Φ and (4.10) for φ = ¯Φ, one has ρ ¯Φ v ( R v h ) = ˆ R n ¯Φ (cid:18) ∞ X k =0 (1 − λ ) λ k M kv (cid:16) h − λ (cid:17)(cid:19) v dx ≤ (1 − λ ) ∞ X k =0 λ k ˆ R n ¯Φ (cid:18) M kv (cid:16) h − λ (cid:17)(cid:19) v dx ≤ (1 − λ ) ∞ X k =0 λ k K k ˆ R n ¯Φ (cid:16) h − λ (cid:17) v dx = 2(1 − λ ) ˆ R n ¯Φ (cid:16) h − λ (cid:17) v dx ≤ ρ ¯Φ v (2 h ) , which proves the second one in (4.11). To proceed, fix a pair ( f, g ) ∈ F with ρ Φ v ( f u ) < ∞ and ρ Φ v ( gu ) < ∞ . We define h ( x ) := Φ( f ( x ) u ( x )) f ( x ) u ( x ) , if f ( x ) u ( x ) = 0 , , if f ( x ) u ( x ) = 0 , x ∈ R n . We may assume that h is non-zero on a set of positive measure. Otherwise, h = 0 a.e., andhence, f u = 0 a.e.. The later implies that f = 0 a.e. since 0 < u < ∞ a.e.. Thus, (4.2) holds.From (4.11) and Lemma 2.1 (2) and (6), we deduce that w := u ( R v h ) v ∈ A ∞ . Fix ε ∈ (0 , r > I Φ . It follows from (2.12) that there is a constant C Φ > ts ) ≤ C Φ t r Φ( s ) for every s > t > . (4.12)This gives that ρ Φ v ( ε − gu ) ≤ C Φ ε − r ρ Φ v ( gu ) . (4.13)Additionally, from the convexity of ¯Φ, (4.11), (2.11) and the definition of h , we obtain that ρ ¯Φ v ( ε R v h ) ≤ ερ ¯Φ v ( R v h ) ≤ ερ ¯Φ v (2 h ) ≤ εCρ ¯Φ v (2 h ) ≤ εCρ Φ v ( f u ) . (4.14)By (4.11) again, one has ρ Φ v ( f u ) = ˆ R n f u h v dx ≤ ˆ R n f u ( R v h ) v dx = k f k L ( w ) . On the other hand, gathering (2.9), (4.13) and (4.14), we deduce that k g k L ( w ) = ˆ R n gu ( R v h ) v dx ≤ ρ ¯Φ v ( ε R v h ) + ρ Φ v ( ε − gu ) ≤ C (cid:0) ερ Φ v ( f u ) + ε − r ρ Φ v ( gu ) (cid:1) . This proves (4.4) and completes the proof. (cid:3)
Theorems 3.5 and 4.1 immediately imply the following result.
Corollary 4.2.
Let Φ( t ) = t or Φ be a Young function with < i Φ ≤ I Φ < ∞ . Suppose that X is a RIBFS over ( R n , dx ) such that q X < ∞ . If for some p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F , (4.15) then for every u ∈ RH ∞ and for every v ∈ A ∞ ∩ L ( R n ) , k Φ( f u ) k X v ≤ C k Φ( gu ) k X v , ( f, g ) ∈ F . (4.16) Recalling (2.8) and (2.18), we have that u := ( M A f ) − λ ∈ RH ∞ for any λ >
0, where A ( t ) = t or A is a Young function. Note that for any invertible function Φ, k Φ( f ) k L , ∞ ( w ) = sup λ> λw ( { x : Φ( f ( x )) > λ } ) = sup λ> Φ( λ ) w ( { x : f ( x ) > λ } ) . From these and Corollary 4.2, we conclude the estimates below.
Corollary 4.3.
Let T be an operator, A ( t ) = t or A be a Young function, and Φ( t ) = t or Φ be Young function such that < i Φ ≤ I Φ < ∞ . Let E ⊂ R n be a measurable set. Suppose that X is a RIBFS over ( R n , dx ) such that q X < ∞ . If for some p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k T f k L p ( E,w ) ≤ C k M A f k L p ( E,w ) . (4.17) then for every v ∈ A ∞ ∩ L ( R n ) , (cid:13)(cid:13)(cid:13)(cid:13) Φ (cid:18) | T f | M A f (cid:19) E (cid:13)(cid:13)(cid:13)(cid:13) X v ≤ C k E k X v . (4.18) In particular, sup λ> Φ( λ ) v ( { x ∈ E : | T f ( x ) | > λM A f ( x ) } ) ≤ Cv ( E ) . Theorem 4.4.
Let Φ be a Young function with < i Φ ≤ I Φ < ∞ . Let u and v be weights on R n such that v ∈ A ∩ L ( R n ) . If for some p ∈ (0 , ∞ ) and for every w ∈ A , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F , (4.19) then ˆ R n Φ( f p u p ) v dx ≤ C ˆ R n Φ( g p u p ) v dx, ( f, g ) ∈ F , (4.20) provided that ˆ R n ¯Φ(( M v f ) u − p ) v dx ≤ K ˆ R n ¯Φ( f u − p ) v dx, ∀ f ∈ M . (4.21) Proof.
As before, it suffices to show that: if (4.21) holds, then for every pair ( f, g ) ∈ F with ρ Φ v ( f p u p ) < ∞ and ρ Φ v ( g p u p ) < ∞ , there exists a weight w = w ( f, g ) ∈ A such that ρ Φ v ( f p u p ) ≤ k f k p L p ( w ) and k g k p L p ( w ) ≤ C (cid:0) ερ Φ v ( f p u p ) + ε − r ρ Φ v ( g p u p ) (cid:1) , (4.22)where ε ∈ (0 ,
1) is an arbitrary number, C > r > ε .To demonstrate (4.22), we fix a pair ( f, g ) ∈ F with ρ Φ v ( f p u p ) < ∞ and ρ Φ v ( g p u p ) < ∞ .Define h ( x ) := Φ( f ( x ) p u ( x ) p ) f ( x ) p u ( x ) p , if f ( x ) u ( x ) = 0 , , if f ( x ) u ( x ) = 0 , x ∈ R n . For the function h , we define R h := ∞ X j =0 M jv h j K j and H := R ( hu p ) u − p . Together with (4.21), a straigtforward calculation gives that h ≤ H, ρ ¯Φ v ( H ) ≤ ρ ¯Φ v (2 h ) and [ R h ] A ( v ) ≤ K . (4.23) WO-WEIGHT EXTRAPOLATION 31
In view of Lemma 2.1 (6) and v ∈ A , we have w := u p Hv = R ( hu p ) v ∈ A . Then the firstestimate in (4.23) gives that ρ Φ v ( f p u p ) = ˆ R n f p u p hv dx ≤ ˆ R n f p u p Hv dµ = k f k p L p ( w ) . In addition, by (2.9), the convexity of Φ and (4.12), we have k g k p L p ( w ) = ˆ R n g p u p Hv dx ≤ ρ ¯Φ v ( εH ) + ρ Φ v ( ε − g p u p ) ≤ ερ ¯Φ v ( H ) + ρ Φ v ( ε − g p u p ) ≤ ερ ¯Φ v (2 h ) + ε − r C Φ ρ Φ v ( g p u p ) ≤ C (cid:0) ερ Φ v ( f p u p ) + ε − r ρ Φ v ( g p u p ) (cid:1) . This proves (4.22). (cid:3)
Theorem 4.5.
Let Φ be a Young function with < i Φ ≤ I Φ < ∞ . Let u and v be weights on R n such that v ∈ A ∞ ∩ L ( R n ) . If for some p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F , (4.24) then for every p ∈ (0 , ∞ ) , ˆ R n Φ( f p u ) v dx ≤ C ˆ R n Φ( g p u ) v dx, ( f, g ) ∈ F , (4.25) provided that ˆ R n ¯Φ(( M v f ) u − ) v dx ≤ K ˆ R n ¯Φ( f u − ) v dx, ∀ f ∈ M . (4.26) Moreover, under the hypothesis (4.26) , (4.24) implies that for every p, q ∈ (0 , ∞ ) , ˆ R n Φ (cid:18)(cid:16) X j f qj (cid:17) q u (cid:19) v dx ≤ C ˆ R n Φ (cid:18)(cid:16) X j g qj (cid:17) q u (cid:19) v dx, { ( f j , g j ) } ⊂ F . (4.27) Proof.
By Theorem B , we have that for every p ∈ (0 , ∞ ) and for every w ∈ A ∞ , k f k L p ( w ) ≤ C k g k L p ( w ) , ( f, g ) ∈ F . (4.28)This can be rewritten as k f p k L ( w ) ≤ C p k g p k L ( w ) , ( f, g ) ∈ F . (4.29)It is enough to show that if (4.26) holds, then for every pair ( f, g ) ∈ F with ρ Φ v ( f u ) < ∞ and ρ Φ v ( gu ) < ∞ , there exists a weight w = w ( f, g ) ∈ A ∞ such that [ w ] A ∞ ≤ K [ v ] A ∞ , ρ Φ v ( f u ) ≤ k f k L ( w ) and k g k L ( w ) ≤ C (cid:0) ερ Φ v ( f u ) + ε − r ρ Φ v ( gu ) (cid:1) , (4.30)where ε ∈ (0 ,
1) is an arbitrary number, C > r > ε . Then (4.25)will follow from (4.25) and (4.29).Indeed, the proof of (4.30) is similar to that of (4.22). We here only present the maindifference. Define R h := ∞ X j =0 M jv h j K j , and H := R ( hu ) u − . Together with (4.26), a straigtforward calculation gives that h ≤ H, ρ ¯Φ v ( H ) ≤ ρ ¯Φ v (2 h ) and [ R h ] A ( v ) ≤ K . (4.31) Since v ∈ A ∞ , we have w := uHv = R ( hu ) v ∈ A ∞ with [ w ] A ∞ ≤ [ R ( hu )] A ( v ) [ v ] A ∞ ≤ K [ v ] A ∞ . Using the same argument as in Theorem 4.4, one can conclude (4.30).Finally, (4.30) is a consequence of (4.25) and (4.28). (cid:3) Extrapolation for commutators
Our next goal is to establish the extrapolation for commutators in the case of two-weight andBanach function spaces. Let us first recall the notion of commutators. Given an operator T and measurable functions b = ( b , . . . , b k ), we define, whenever it makes sense, the commutatorby C b ( T )( f )( x ) := T (cid:18) k Y j =1 ( b j ( x ) − b j ( · )) f ( · ) (cid:19) ( x ) . When b = ( b, . . . , b ), we denote C kb ( T ) instead of C b ( T ).We say that a sublinear operator T is linearizable if there exists a Banach space B anda B -valued linear operator T such that T f ( x ) = kT f ( x ) k B . In this way, we set C b ( T ) f ( x ) := k C b ( T ) f ( x ) k B = (cid:13)(cid:13)(cid:13)(cid:13) T (cid:16) k Y j =1 (cid:0) b j ( x ) − b j ( · ) (cid:1) f ( · ) (cid:17) ( x ) (cid:13)(cid:13)(cid:13)(cid:13) B . Given a weight ν , we say that a locally integrable function f ∈ BMO( ν ) if k f k BMO( ν ) := sup Q ν ( Q ) ˆ Q | f ( x ) − f Q | dx < ∞ . If ν ≡
1, we simply write BMO := BMO( ν ). Define a new Orlicz-type space BMO := BMO ( R n ) via the norm k f k BMO := sup Q k f − f Q k exp L,Q . The John-Nirenberg inequality says that there exists a dimensional constant C n such thatfor every f ∈ BMO, k f k BMO ≤ k f k BMO ≤ C n k f k BMO . (5.1)Thus, (5.1) implies that (BMO , k · k BMO ) and (
BMO , k · k BMO ) are equivalent quasi-normedspaces; nevertheless, the appearance of
BMO in the statements of our results below is toemphasize that k · k
BMO is used.It is well-known that if w ∈ A ∞ , then log w ∈ BMO . The converse can be formulated asfollows. Let b ∈ BMO . Then for every 1 ≤ p < ∞ , λ ∈ R : | λ | ≤ min { , p − }k b k − BMO = ⇒ [ e λb ] A p ≤ | λ |k b k BMO . (5.2)Let us recall the sharp reverse H¨older’s inequality from [17, 42, 48]. For every w ∈ A p with1 ≤ p ≤ ∞ , (cid:18) Q w r w dx (cid:19) rw ≤ Q w dx, (5.3) WO-WEIGHT EXTRAPOLATION 33 for every cube Q , where r w = n +1 [ w ] A , p = 1 , n +1+2 p [ w ] Ap , p ∈ (1 , ∞ ) , n +11 [ w ] A ∞ , p = ∞ . Now we state the main theorem of this section.
Theorem 5.1.
Fix < s i , θ i < ∞ for i = 1 , , , . Let σ and ν be weights on R n , X σ and Y σ be Banach function spaces over ( R n , σ ) . Let T be either a linear or a linearizable operator.Assume that for every pair of weights ( µ, λ ) with ν = µ/λ such that ( µ θ , λ θ , µ − θ , λ − θ ) ∈ A s × A s × A s × A s , k ( T f ) λ k Y σ ≤ Ψ (cid:16) [ µ θ ] A s , [ λ θ ] A s , [ µ − θ ] A s , [ λ − θ ] A s (cid:17) k f µ k X σ , (5.4) where Ψ : [1 , ∞ ) × [1 , ∞ ) × [1 , ∞ ) × [1 , ∞ ) → [1 , ∞ ) is an increasing function of each variable.Then for every k ≥ , for every b = ( b , . . . , b k ) ∈ BMO k and for every pair of weights ( µ, λ ) with ν = µ/λ such that ( µ θ , λ θ , µ − θ , λ − θ ) ∈ A s × A s × A s × A s , k ( C b ( T ) f ) λ k Y σ ≤ n +2+2 max { s i ,s ′ i } θk min { , s i − } ! k max (cid:8) [ µ θ ] r A s , [ λ θ ] r A s , [ µ − θ ] r A s , [ λ − θ ] r A s (cid:9) k × Ψ (cid:16) t [ µ θ ] A s , t [ λ θ ] A s , t [ µ − θ ] A s , t [ λ − θ ] A s (cid:17) k Y j =1 k b j k BMO k f µ k X σ , where θ = max { θ , θ , θ , θ } , r i = max { , s i − } and t i := s i + 2 min { , s i − } .Proof. We will borrow the idea from [10], where the extrapolation for commutators was ob-tained in the one-weight case and in Lebesgue spaces. Let u := µ θ ∈ A s . Then v := u − s ′ ∈ A s ′ . Due to (5.3), one has (cid:18) Q u r u dx (cid:19) ru ≤ Q u dx and (cid:18) Q v r v dx (cid:19) rv ≤ Q v dx (5.5)for every cube Q , where r u = 1 + 12 n +1+2 s [ u ] A s and r v = 1 + 12 n +1+2 s ′ [ v ] A s ′ (5.6)Set η := min { r u , r v } . Taking into account (5.5) and (5.6), we obtain[ µ θ η ] η A s ≤ s [ µ θ ] A s with η ′ = 1 + 2 n +1+2 max { s ,s ′ } [ µ θ ] max { , s − } A s . (5.7)Analogously,[ λ θ η ] η A s ≤ s [ λ θ ] A s with η ′ = 1 + 2 n +1+2 max { s ,s ′ } [ λ θ ] max { , s − } A s , [ µ − θ η ] η A s ≤ s [ µ − θ ] A s with η ′ = 1 + 2 n +1+2 max { s ,s ′ } [ λ − θ ] max { , s − } A s , [ λ − θ η ] η A s ≤ s [ λ − θ ] A s with η ′ = 1 + 2 n +1+2 max { s ,s ′ } [ λ − θ ] max { , s − } A s . To continue, we writeΦ z f ( x ) := e z b ( x )+ ··· + z k b k ( x ) T (cid:0) e − z b −···− z k b k f (cid:1) ( x ) , z = ( z . . . , z k ) ∈ C k . By Cauchy integral formula adapted to several complex variables, we get C b ( T ) f ( x ) = ∂ z Φ z f ( x ) (cid:12)(cid:12) z =0 = 1(2 πi ) k ˆ ∂P δ Φ z f ( x ) z · · · z k dz · · · dz k , (5.8)where P δ = { z = ( z , . . . , z k ) ∈ C k : | z j | < δ j , j = 1 , . . . , k } . Pick δ j := min { , s − , s − , s − , s − , } k max { θ η ′ , θ η ′ , θ η ′ , θ η ′ }k b j k BMO , j = 1 , . . . , k. (5.9)Fix z ∈ P δ , denote U ( x ) := µ ( x ) W ( x ) , V ( x ) := λ ( x ) W ( x ) and W ( x ) := e Re( z ) b ( x )+ ··· +Re( z k ) b k ( x ) . Then it follows from H¨older’s inequality and (5.2) that (cid:18) Q W θ η ′ dx (cid:19)(cid:18) Q W θ η ′ (1 − s ′ ) dx (cid:19) s − ≤ k Y j =1 (cid:18) Q e kθ η ′ Re( z j ) b j dx (cid:19) k (cid:18) Q e kθ η ′ Re( z j ) b j (1 − s ′ ) dx (cid:19) s − k ≤ k Y j =1 θ η ′ | Re( z j ) |k b j k BMO ≤ k Y j =1 min { ,s − } k = 4 min { ,s − } . Together with H¨older’s inequality and (5.7), this in turn gives (cid:18) Q U θ dx (cid:19)(cid:18) Q U θ (1 − s ′ ) dx (cid:19) s − ≤ (cid:18) Q µ θ η dx (cid:19) η (cid:18) Q W θ η ′ dx (cid:19) η ′ × (cid:18) Q µ θ η (1 − s ′ ) dx (cid:19) s − η (cid:18) Q W θ η ′ µ (1 − s ′ ) dx (cid:19) s − η ′ ≤ min { ,s − } [ µ θ η ] η A s ≤ min { ,s − } s [ µ θ ] A s . That is, [ U θ ] A s ≤ min { ,s − } s [ µ θ ] A s . (5.10)Similarly, [ V θ ] A s ≤ min { ,s − } s [ λ θ ] A s , (5.11)[ U − θ ] A s ≤ min { ,s − } s [ µ − θ ] A s , (5.12)[ V − θ ] A s ≤ min { ,s − } s [ λ − θ ] A s . (5.13) WO-WEIGHT EXTRAPOLATION 35
Note that
U/V = µ/λ = ν . This together with (5.10)–(5.13) and (5.4) gives that k (Φ z f ) λ k Y σ = k T ( e − z b −···− z m b m f ) V k Y σ ≤ Ψ (cid:16) [ U θ ] A s , [ V θ ] A s , [ U − θ ] A s , [ V − θ ] A s (cid:17) k ( W − f ) U k X σ ≤ Ψ (cid:16) t [ µ θ ] A s , t [ λ θ ] A s , t [ µ − θ ] A s , t [ λ − θ ] A s (cid:17) k f µ k X σ . (5.14)Now fix a non-negative function h ∈ Y ′ σ with k h k Y ′ σ ≤
1. Invoking (5.8) and H¨older’sinequality (2.30), we obtain ˆ R n | C b ( T ) f ( x ) | λ ( x ) h ( x ) v ( x ) dx ≤ ˆ R n (cid:18) π ) k ˆ ∂P δ | Φ z f ( x ) || z | · · · | z k | | dz | · · · | dz k | (cid:19) λ ( x ) h ( x ) v ( x ) dx ≤ k Y j =1 δ − j π ) k ˆ ∂P δ (cid:18) ˆ R n | Φ z f ( x ) | λ ( x ) h ( x ) v ( x ) dx (cid:19) | dz | · · · | dz k |≤ k Y j =1 δ − j π ) k ˆ ∂P δ k (Φ z f ) λ k Y σ k h k Y ′ σ | dz | · · · | dz k | . (5.15)Then, combining (2.31), (5.14) and (5.15), we conclude that k ( C b ( T ) f ) λ k Y σ ≤ k Y j =1 δ − j π ) k ˆ ∂P δ k (Φ z f ) λ k Y σ | dz | · · · | dz k | . ≤ k Y j =1 δ − j Ψ (cid:16) t [ µ θ ] A s , t [ λ θ ] A s , t [ µ − θ ] A s , t [ λ − θ ] A s (cid:17) k f µ k X σ . Therefore, the desired estimate follows at once this and (5.9). (cid:3)
Recall that a family S of cubes is called sparse if for every cube Q ∈ S , there exists E Q ⊂ Q such that | E Q | ≥ η | Q | for some 0 < η < { E Q } Q ∈S is pairwise disjoint.Given a sparse family S and γ ≥
1, we define a sparse operator as A γ S ( f )( x ) := (cid:16) X Q ∈S h f i γQ Q ( x ) (cid:17) /γ , x ∈ R n where h f i Q = | Q | ´ Q f dx . If γ = 1, we denote A S = A γ S . Corollary 5.2.
If for every f, g ∈ C ∞ c ( R n ) and for every b ∈ L ( R n ) , |h C b ( T ) f, g i| . sup S is sparse ( T S ( b, f, g ) + T ∗S ( b, f, g )) , (5.16) where T S ( b, f, g ) := X Q ∈S h| f |i Q h| ( b − b Q ) g |i Q | Q | , T ∗S ( b, f, g ) := X Q ∈S h| ( b − b Q ) f |i Q h| g |i Q | Q | , then for every p ∈ (1 , ∞ ) , for every µ p , λ p ∈ A p with ν = µ/λ and for every b = ( b , b , . . . , b k ) ∈ BMO( ν ) × BMO × · · · ×
BMO , k C b ( T ) k L p ( µ p ) → L p ( λ p ) . k b k BMO( ν ) k Y j =2 k b j k BMO (cid:0) [ µ p ] A p [ λ p ] A p (cid:1) max { , p − } × max (cid:8) [ µ p ] A p , [ λ p ] A p (cid:9) ( k −
1) max { , p − } . (5.17) Proof.
Fix p ∈ (1 , ∞ ) and µ p , λ p ∈ A p with ν = µ/λ . Let S be a sparse family, b := b ∈ BMO( ν ) and b j ∈ BMO, j = 2 , . . . , k . It is well-known that kA S k L p ( w ) → L p ( w ) ≤ c n,p [ w ] max { , p − } A p , ∀ p ∈ (1 , ∞ ) . (5.18)Recently, Lerner et al. [51, Lemma 5.1] proved that for a sparse family S , there exists anothersparse family e S ⊂ D containing S and such that for every Q ∈ e S , | b ( x ) − b Q | ≤ n +2 X Q ′ ∈ e S : Q ′ ⊂ Q h| b − b Q ′ |i Q ′ Q ′ ( x ) , a.e. x ∈ Q. This immediately gives that h| ( b − b Q ) g |i Q . k b k BMO( ν ) hA e S ( | g | ) ν i Q . (5.19)Thus, using (5.19), H¨older’s inequality, (2.5) and (5.18), we deduce that T S ( b, f, g ) . k b k BMO( ν ) X Q ∈S h| f |i Q hA e S ( | g | ) ν i Q | Q | . k b k BMO( ν ) X Q ∈S (cid:16) inf Q M f (cid:17)(cid:16) inf Q M ( A e S ( | g | ) ν ) (cid:17) | E Q |≤ k b k BMO( ν ) k M f · M ( A e S ( | g | ) ν ) k L ( R n ) ≤ k b k BMO( ν ) k M f k L p ( µ p ) k M ( A e S ( | g | ) ν ) k L p ′ ( µ − p ′ ) . k b k BMO( ν ) [ µ p ] p − A p [ µ − p ′ ] p ′− A p ′ k f k L p ( µ p ) kA e S ( | g | ) k L p ′ ( λ − p ′ ) . k b k BMO( ν ) (cid:0) [ µ p ] A p [ λ p ] A p (cid:1) p − (cid:0) [ µ − p ′ ] A p ′ [ λ − p ′ ] A p ′ (cid:1) p ′− k f k L p ( µ p ) k g k L p ′ ( λ − p ′ ) . Similarly, one has T ∗S ( b, f, g ) . k b k BMO( ν ) k M ( A e S ( | f | ) ν ) · M g k L ( R n ) . k b k BMO( ν ) [ λ p ] p − A p [ µ p ] max { , p − } A p [ λ − p ′ ] p ′− A p ′ k f k L p ( µ p ) k g k L p ′ ( λ − p ′ ) . k b k BMO( ν ) (cid:0) [ µ p ] A p [ λ p ] A p (cid:1) p − (cid:0) [ µ − p ′ ] A p ′ [ λ − p ′ ] A p ′ (cid:1) p ′− k f k L p ( µ p ) k g k L p ′ ( λ − p ′ ) . Collecting the above estimates and (5.16), we conclude that k C b ( T ) k L p ( µ p ) → L p ( λ p ) . k b k BMO( ν ) (cid:0) [ µ p ] A p [ λ p ] A p (cid:1) p − (cid:0) [ µ − p ′ ] A p ′ [ λ − p ′ ] A p ′ (cid:1) p ′− . (5.20)Consequently, (5.17) follows at once from (5.1) and Theorem 5.1 applied to the case X σ = Y σ = L p ( R n ), s = s = θ = θ = p , s = s = θ = θ = p ′ , andΨ( t , t , t , t ) = k b k BMO( ν ) ( t t ) p − ( t t ) p ′− . The proof is complete. (cid:3)
WO-WEIGHT EXTRAPOLATION 37
Theorem 5.3.
For every p, q ∈ (1 , ∞ ) , for every µ, λ ∈ A p , for every b = ( b , b , . . . , b k ) ∈ BMO( ν ) × BMO × · · · ×
BMO with ν = ( µ/λ ) /p , k T k L p ( µ ) → L p ( λ ) . k b k BMO( ν ) k Y j =2 k b j k BMO (cid:0) [ µ ] A p [ λ ] A p (cid:1) max { , p − } × max (cid:8) [ µ ] A p , [ λ ] A p (cid:9) ( k −
1) max { , p − } , for every operator T ∈ (cid:8) C b ( T ) , | C b ( T ) | q , V ρ ◦ C b ( T ) , C b ( T σ ) (cid:9) , where T ∈ ω -CZO with ω ∈ Dini . The definitions of operators above can be found in Section 6. Let us see how Corollary5.2 implies Theorem 5.3. Indeed, it is enough to show that each operator T above verifies(5.16). This will follow from the point sparse domination obtained in [51, Theorem 1.4], [16,Theorem 1.11], [70, Proposition 3.3] and [14, Proposition 4.1] respectively.Next, we apply Theorem 5.1 to establish Bloom type inequality for the fractional integraloperator I α (0 < α < n ) given by I α f ( x ) := ˆ R n f ( y ) | x − y | n − α dy, x ∈ R n . Let us see the first order commutator. It was proved in [1] that k C b ( I α ) k L p ( µ p ) → L q ( λ q ) . Ξ( µ, λ ) k b k BMO( ν ) , where Ξ( µ, λ ) = [ µ ] (1 − αn ) max { , p ′ q } A p,q [ λ q ] max { , q − } A q + [ λ ] (1 − αn ) max { , p ′ q } A p,q [ µ p ] max { , p − } A p . In view of (2.3) and (2.4), we rewrite Ξ( µ, λ ) asΞ( µ, λ ) = Ψ (cid:16) [ µ p ] A p , [ λ q ] A qp ′ , [ µ − p ′ ] A p ′ q , λ − q ′ ] A q ′ (cid:17) , where Ψ( t , t , t , t ) := t max { , p − } t (1 − αn ) max { , p ′ q } + t (1 − αn ) max { , qp ′ } t max { , q ′− } . (5.21)Therefore, the result below immediately follows from Theorem 5.1 applied to the case X σ = L p ( R n ), Y σ = L q ( R n ), ( s , s , s , s ) = ( p, qp ′ , p ′ q , q ′ ), ( θ , θ , θ , θ ) = ( p, q, p ′ , q ′ ), andthe function Ψ defined in (5.21). Theorem 5.4.
Let < α < n and < p < q < ∞ with p − q = αn . Then for every µ, λ ∈ A p,q and for every b = ( b , b , . . . , b k ) ∈ BMO( ν ) × BMO × · · · ×
BMO with ν = µ/λ , k C b ( I α ) k L p ( µ p ) → L q ( λ q ) . k b k BMO( ν ) k Y j =2 k b j k BMO Ξ( µ, λ ) × max n [ µ ] max { , p ′ q } A p,q , [ µ p ] max { , p − } A p , [ λ ] max { , p ′ q } A p,q , [ λ q ] max { , q − } A q o k − . Applications
The purpose of this section is to present some applications of the extrapolation theoremsestablished above. We will see that by means of extrapolation, many known results can beextended to the general Banach function spaces. Before doing that, we recall the definitionsand notation of some operators.Let ω : [0 , → [0 , ∞ ) be a modulus of continuity, that is, ω is increasing, subadditive and ω (0) = 0. We say that T is an ω -Calder´on-Zygmund operator (or simply, T ∈ ω -CZO), if T is L bounded and represented as T f ( x ) = ˆ R n K ( x, y ) f ( y ) dx, ∀ x supp( f ) , where the kernel K : R n × R n \ { ( x, y ) ∈ R n × R n : x = y } → C is a function satisfying thefollowing conditions: • Size condition: | K ( x, y ) | . | x − y | n , ∀ x = y , • Smoothness conditions: | K ( x, y ) − K ( x ′ , y ) | . ω (cid:18) | x − x ′ || x − y | (cid:19) | x − y | n , whenever | x − x ′ | ≤ | x − y | , | K ( x, y ) − K ( x, y ′ ) | . ω (cid:18) | y − y ′ || x − y | (cid:19) | x − y | n , whenever | y − y ′ | ≤ | x − y | . Throughout this section, whenever T ∈ ω -CZO, we always assume that ω satisfies the Dinicondition (or, ω ∈ Dini), which means that k ω k Dini := ´ ω ( t ) dtt < ∞ . An example of Dinicondition is ω ( t ) = t α with α >
0. In this case, we say that T is a standard Calder´on-Zygmundoperator , or T ∈ α -CZO.Given an ω -Calder´on-Zygmund operator T , we define its truncated singular integral by T ǫ f ( x ) := ˆ | x − y | >ǫ K ( x, y ) f ( y ) dy, Then for ρ >
2, the ρ -variation operator for the families of operators T := { T ǫ } ǫ> and T b := C b ( T ) = { T b,ǫ = C b ( T ǫ ) } ǫ> are defined as V ρ ( T f )( x ) := sup { ǫ j }↓ (cid:18) ∞ X j =1 | T ǫ j +1 f ( x ) − T ǫ j f ( x ) | ρ (cid:19) ρ , (6.1) V ρ ( T b f )( x ) := sup { ǫ j }↓ (cid:18) ∞ X j =1 | T b,ǫ j +1 f ( x ) − T b,ǫ j f ( x ) | ρ (cid:19) ρ , (6.2)where the supremum is taken over all sequences { ǫ j } decreasing to zero.Given a symbol σ , the pseudo-differential operator T σ is defined by T σ f ( x ) = ˆ R n σ ( x, ξ ) e πix · ξ b f ( ξ ) dξ, (6.3)where the Fourier transform b f of the function f . Given m ∈ R and ̺, δ ∈ [0 , σ on R n × R n belongs to H¨ormander class S m̺,δ if for each triple of multi-indices α and β there exists a constant C α,β such that (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ σ ( x, ξ ) (cid:12)(cid:12)(cid:12) ≤ C α,β (1 + | ξ | ) m − ρ | β | + δ | α | . WO-WEIGHT EXTRAPOLATION 39
Let us introduce two kinds of more singular operators. Recall that the rough singularintegral T Ω is defined by T Ω f ( x ) := p . v . ˆ R n Ω( y ′ ) | y | n f ( x − y ) dy, (6.4)where Ω ∈ L ∞ ( S n − ) and ´ S n − Ω( ξ ) dσ ( ξ ) = 0. On the other hand, the Bochner-Rieszmultiplier is defined by d B δ f ( ξ ) := (1 − | ξ | ) δ + b f ( ξ ) . (6.5)Next, we introduce a class of square functions. A function K defined on R n × R n is said tobe in LP if there exist β > γ > • Size condition: | K ( x, y ) | . | x − y | ) n + β , ∀ x, y ∈ R n , • Smoothness conditions: | K ( x, y ) − K ( x ′ , y ) | . | x − x ′ | γ (1 + | x − y | ) n + β + γ , whenever | x − x ′ | ≤ | x − y | , | K ( x, y ) − K ( x, y ′ ) | . | y − y ′ | γ (1 + | x − y | ) n + β + γ , whenever | y − y ′ | ≤ | x − y | . Given a function K ∈ LP, we always denote K t f ( x ) := 1 t n ˆ R n K (cid:16) xt , yt (cid:17) f ( y ) dy . Then for α ≥ λ >
2, we define the square functions as S α ( f )( x ) := (cid:18) ¨ Γ α ( x ) | K t f ( x ) | dydtt n +1 (cid:19) , (6.6) g ∗ λ ( f )( x ) := (cid:18) ¨ R n +1+ (cid:16) tt + | x − y | (cid:17) nλ | K t f ( y ) | dydtt n +1 (cid:19) , (6.7)where Γ α ( x ) = { ( y, t ) ∈ R n +1+ : | x − y | < αt } .6.1. Local decay estimates.
In this subsection, let us see how to use Theorem 3.1 to establishlocal decay estimates: ψ t ( T , M ) := sup Q : cube ⊂ R n sup f ∈ L ∞ c ( R n )supp( f ) ⊂ Q | Q | − |{ x ∈ Q : | T f ( x ) | > t M f ( x ) }| , t > , where T is a singular operator and M is an appropriate maximal operator. For several kindof singular operators, we will obtain the exponential decay by means of the method of sparsedomination. This kind of estimates originated in the work of Karagulyan [44] and was furtherinvestigated by Ortiz-Caraballo, P´erez and Rela [61] using a novel strategy.Let T be the ω -Calder´on-Zygmund operator with ω ∈ Dini. Recall that the ρ -variationoperators V ρ ◦ T and V ρ ◦ T b are defined in (6.1) and (6.15). Fix a cube Q and f ∈ L ∞ c ( R n )with supp( f ) ⊂ Q . With these notation in hand, we have for every 1 ≤ q < ∞ and for every w ∈ A q , k T f k L ( Q,w ) ≤ c n,q [ w ] A q k M f k L ( Q,w ) , (6.8) kV ρ ( T f ) k L ( Q,w ) ≤ c n,q [ w ] A q k M f k L ( Q,w ) , (6.9) k T σ f k L ( Q,w ) ≤ c n,q [ w ] A q k M f k L ( Q,w ) , (6.10) k S α f k L ( Q,w ) ≤ c n,q [ w ] A q k M f k L ( Q,w ) , (6.11) k g ∗ λ f k L ( Q,w ) ≤ c n,q [ w ] A q k M f k L ( Q,w ) , (6.12) k T Ω f k L ( Q,w ) ≤ c n,q [ w ] A q k M f k L ( Q,w ) , (6.13) k B ( n − / f k L ( Q,w ) ≤ c n,q [ w ] A q k M f k L ( Q,w ) , (6.14) kV ρ ( T b f ) k L ( Q,w ) ≤ c n,q k b k BMO [ w ] A q k M f k L ( Q,w ) , (6.15) k C b ( T σ ) f k L ( Q,w ) ≤ c n,q k b k BMO [ w ] A q k M f k L ( Q,w ) , (6.16) k C b ( T Ω ) f k L ( Q,w ) ≤ c n,q k b k BMO [ w ] A q k M f k L ( Q,w ) , (6.17) k C kb ( T ) f k L ( Q,w ) ≤ c n,q k b k k BMO [ w ] k +1 A q k M k +1 f k L ( Q,w ) , (6.18)where c n,q is independent of Q , f and w . Indeed, using the techniques in [14, 61], one canshow (6.8)–(6.12). But it needs the sharp maximal function control for these operators. Thisstrategy is invalid for T Ω and B ( n − / . To circumvent this problem, we present a local versionof sparse domination for all the operators above. We only give the proof of (6.17) since theother proofs are similar and simpler.We only consider the case w ∈ A q with 1 < q < ∞ . Recall that f ∈ L ∞ c ( R n ) withsupp( f ) ⊂ Q . Modifying the proof of [64, Theorem 3.1], we obtain that for every g ∈ L ∞ c ( R n ),for every b ∈ L ( R n ) and for every s ∈ (1 , ∞ ), there exists a sparse family S ( Q ) ⊂ D ( Q ) suchthat |h C b ( T Ω ) f, g Q i| . s ′ (cid:0) Λ Q,s ( b, f, g ) + Λ ∗ Q,s ( b, f, g ) (cid:1) , (6.19)where Λ Q,s ( b, f, g ) := X Q ′ ∈S ( Q ) h| f |i Q ′ h| ( b − b Q ′ ) g |i Q ′ ,s | Q ′ | , Λ ∗ Q,s ( b, f, g ) := X Q ′ ∈S ( Q ) h| ( b − b Q ′ ) f |i Q ′ h| g |i Q ′ ,s | Q ′ | . By w ∈ A q and (5.3), one has w ∈ RH r w , where r w = 1 + n +1+2 q [ w ] Aq . If we set r =1 + n +2+2 q [ w ] Aq and s = r w r , then 1 < s < s ′ ≃ [ w ] A q ≃ r ′ . A useful inequality from[33, Corollary 3.1.8] is that k b − b Q ′ k p,Q ′ ≤ c n p k b k BMO for all p >
1. Using these estimates, wehave Λ
Q,s ( b, f, gw ) = X Q ′ ∈S ( Q ) h| f |i Q ′ h| ( b − b Q ′ ) gw |i Q ′ ,s | Q ′ |≤ X Q ′ ∈S ( Q ) h| f |i Q ′ h| b − b Q ′ |i Q ′ ,sr ′ h w i Q ′ ,sr k g k L ∞ ( R n ) | Q ′ | . X Q ′ ∈S ( Q ) h| f |i Q ′ h| b − b Q ′ |i Q ′ ,sr ′ h w i Q ′ ,sr k g k L ∞ ( R n ) | Q ′ | WO-WEIGHT EXTRAPOLATION 41 . sr ′ k b k BMO k g k L ∞ ( R n ) X Q ′ ∈S ( Q ) h| f |i Q ′ w ( Q ′ ) . [ w ] A q k b k BMO k g k L ∞ ( R n ) X Q ′ ∈S ( Q ) (cid:18) Q ′ ( M f ) Q dw (cid:19) w ( Q ′ ) . [ w ] A q k b k BMO k M f k L ( Q,w ) k g k L ∞ ( R n ) , (6.20)where we used the Carleson embedding theorem from [43, Theorem 4.5] and that the collection { w ( Q ′ ) } Q ′ ∈S ( Q ) satisfies the Carleson packing condition with the constant c n [ w ] A q . Likewise,one has Λ ∗ Q,s ( b, f, gw ) = X Q ′ ∈S ( Q ) h| ( b − b Q ′ ) f |i Q ′ h| gw |i Q ′ ,s | Q |≤ X Q ′ ∈S ( Q ) k b − b Q ′ k exp L, Q ′ k f k L (log L ) , Q ′ h w i s k g k L ∞ ( R n ) | Q ′ | . k b k BMO k g k L ∞ ( R n ) X Q ′ ∈S ( Q ) k f k L (log L ) , Q ′ w ( Q ′ ) . [ w ] A q k b k BMO k M L (log L ) f k L ( Q,w ) k g k L ∞ ( R n ) . (6.21)Hence, (6.17) follows at once from (6.19), (6.20), (6.21) and that s ′ ≃ [ w ] A q .Let us turn our attention to the local decay estimates. Theorem 6.1.
Let A ( t ) = t or A be a Young function and let E ⊂ R n be a measurable set.Suppose that X is a RIBFS over ( R n , dx ) with q X < ∞ . If for some q ∈ (2 , ∞ ) and for every w ∈ A q , k T f k L ( E,w ) ≤ Ψ (cid:0) [ w ] A q (cid:1) k M A f k L ( E,w ) , (6.22) where Ψ : [1 , ∞ ) → [1 , ∞ ) is an increasing function, then for every v ∈ A ∩ L ( R n ) , (cid:13)(cid:13)(cid:13)(cid:13) T fM A f E (cid:13)(cid:13)(cid:13)(cid:13) X v ≤ (cid:0) c n,q k M v k X ′ v → X ′ v [ v ] A (cid:1) k E k X v . (6.23) In particular, for every p ∈ (1 , ∞ ) , sup t> t |{ x ∈ E : | T f ( x ) | > tM A f ( x ) }| p ≤ C n,q p ) | E | p . (6.24) Proof.
Fix q >
2. Let v ∈ A ∩ L ( R n ) and E ⊂ R n be a measurable set. Set u := ( M A f ) − .By (2.7) and (2.17), one has [ u − q ′ ] A = [( M A f ) q − ] A ≤ C n,q . Then it follows from (6.22)that for every w ∈ A q , k T f · E k L ( w ) = k T f k L ( E,w ) ≤ Ψ (cid:0) [ w ] A q (cid:1) k M A f k L ( E,w ) = k M A f · E k L ( w ) . This verifies (3.1). Hence, Theorem 3.1 applied to the pair (
T f · E , M A f · E ) gives (6.23)as desired. Furthermore, observe that for every p ∈ (1 , ∞ ), k M k L p ′ , ∞ ( R n ) → L p ′ , ∞ ( R n ) . p (see[32, Exercise 2.1.13] or (6.44) below). Therefore, (6.24) is a consequence of (6.23) for the case v ≡ X v = L p, ∞ ( R n ). (cid:3) To proceed, fix p ∈ (1 , ∞ ) chosen later and t >
0. In view of (6.18), the hypothesis (6.22)with Ψ( t ) = c n k b k k BMO t k +1 is verified. Thus, the inequality (6.24) gives that ψ t ( C kb ( T ) , M k +1 ) ≤ (cid:16) c n,k k b k k BMO t − p k +1 (cid:17) p . (6.25) If t > t := c n,k e k b k k BMO , pick p ∈ (1 , ∞ ) such that t = c n,k e k b k k BMO p k +1 . If we denote α = ( c n,k e ) − , then it follows from (6.25) that ψ t ( C kb ( T ) , M k +1 ) ≤ e − p = 2 e − ( αt/ k b k k BMO ) k +1 . (6.26)If 0 < t < t , then p ∈ (1 , ∞ ) can be chosen as an arbitrary number and by definition, ψ t ( C kb ( T ) , M k +1 ) ≤ e · e − ( αt / k b k k BMO ) k +1 ≤ e · e − ( αt/ k b k k BMO ) k +1 . (6.27)Summing (6.26) and (6.27) up, we obtain ψ t ( C kb ( T ) , M k +1 ) . e − ( αt/ k b k k BMO ) k +1 , ∀ k ≥ . A similar argument yields that ψ t ( T, M ) . e − αt , ψ t ( S α , M ) . e − αt , ψ t ( B ( n − / , M ) . e − αt ,ψ t ( V ρ ◦ T, M ) . e − αt , ψ t ( g ∗ λ , M ) . e − αt , ψ t ( V ρ ◦ T b , M ) . e ( − αt/ k b k BMO ) ,ψ t ( T σ , M ) . e − αt , ψ t ( T Ω , M ) . e − αt , ψ t ( C b ( T σ ) , M ) . e ( − αt/ k b k BMO ) ,ψ t ( C b ( T Ω ) , M ) . e ( − αt/ k b k BMO ) . Coifman-Fefferman inequalities.
To simplify notation, we set T ∈ { T, V ρ ◦ T , T σ , T Ω , B ( n − / } , T ∈ { C b ( T ) , V ρ ◦ T b , C b ( T σ ) , C b ( T Ω ) } , (6.28)where T is an ω -Calder´on-Zygmund operator with ω ∈ Dini.
Theorem 6.2.
Let Φ( t ) = t or Φ be a Young function with < i Φ ≤ I Φ < ∞ . Supposethat X is a RIBFS over ( R n , dx ) such that q X < ∞ . Then for every u ∈ RH ∞ and for every v ∈ A ∞ ∩ L ( R n ) , k Φ( T i f · u ) k X v . k Φ( M i f · u ) k X v , i = 1 , . (6.29) Proof.
By Corollary 4.2, the estimate (6.29) follows from the following k T i f k L p ( w ) . k M i f k L p ( w ) , ∀ p ∈ (0 , ∞ ) , ∀ w ∈ A ∞ , i = 1 , , (6.30)where the implicit constants only depend on n , p and [ w ] A ∞ . Hence, it suffices to verify (6.30).By means of sparse dominations aforementioned for T and T , a standard argument will derive(6.30) for p = 1. Then using Theorem B in Section 3, we obtain (6.30) for all p ∈ (0 , ∞ ). (cid:3) Remark 6.3. By (2.8) , we see that u := ( M w ) − µ/p ∈ RH ∞ for all < p < ∞ and µ > .Then taking v ≡ and X = L p ( R n ) , we derive the result for the α -CZO in [46, Theorem 1.7] .If u ≡ and v ∈ A ∞ , then Theorem 6.2 applied to X = L p ( R n ) and X = L p, ∞ ( R n ) yieldsthe weighted inequalities in [54, Corollary 1.2] . By the same reason, Theorem 6.2 gives theestimates that coincide with [16, Theorems 1.9, 1.10] for ω -CZO T and its commutator in thecase w ∈ A ∞ . We are going to present another type of Coifman-Fefferman inequalities. Note that byCoifmann and Rochberg theorem (2.7), we get v := M r w ∈ A for every weight w and every r >
1. Thus, invoking Theorem 3.6 applied to u = v = M r w , Theorem 3.7 applied to u = v = M A p w , (2.18) and (6.30), we obtain the weighted estimates below. WO-WEIGHT EXTRAPOLATION 43
Theorem 6.4.
Let X be a RIBFS over ( R n , dx ) with q X < ∞ . Then for every weight w andfor every r > , (cid:13)(cid:13)(cid:13)(cid:13) T i fM r w (cid:13)(cid:13)(cid:13)(cid:13) X ( M r w ) . (cid:13)(cid:13)(cid:13)(cid:13) M i fM r w (cid:13)(cid:13)(cid:13)(cid:13) X ( M r w ) , i = 1 , . (6.31) Theorem 6.5.
Let A be a Young function and w be a weight. Write A p ( t ) = A ( t /p ) for < p < ∞ . Suppose that X v be a BFS over ( R n , v dx ) with v = M A p w . If there exists r > such that M ′ v /r is bounded on X ′ v , then (cid:13)(cid:13)(cid:13)(cid:13) T i fv (cid:13)(cid:13)(cid:13)(cid:13) X v . (cid:13)(cid:13)(cid:13)(cid:13) M i fv (cid:13)(cid:13)(cid:13)(cid:13) X v , i = 1 , . (6.32)We here mention that (6.31) and (6.32) are key ingredients leading to the sharp A inequal-ities as follows: k T f k L p ( w ) ≤ c n,p,T [ w ] βA k f k L p ( w ) , < p < ∞ , (6.33) k T f k L , ∞ ( w ) ≤ c n,T [ w ] γA log( e + [ w ] A ∞ ) k f k L ( w ) . (6.34)Such estimates originated from [49] and were extensively extended to other singular operatorsand commutators.(1) If X = L p ( R n ) with p > T is a α -CZO, (6.31) was given in [49, Lemma 2.1]. Inthis case, the estimate (6.33) with β = 1 is sharp with respect to [ w ] A . The inequality(6.34) holds for γ = 1 as well.(2) If X = L p ′ ( R n ) with p > T is a α -CZO, (6.31) for T ∗ was obtained in the proofof [60, Theorem 1.3]. Also, (6.33) holds for C b ( T ) and β = 2.(3) If X = L p ′ ( R n ) with p > T = T Ω , (6.31) was established in the proof of [64,Theorem 1.1]. A more accurate bound is obtained and leads to (6.33) with β = 2.(4) If X v = L p ′ ( v ) with p > T ∈ { T Ω , B ( n − / } , (6.32) was got in [54, p.2546]. Arefined endpoint inequality implies (6.34) with γ = 2.(5) The inequality (6.32) also holds for the sparse dyadic operator A S . A particular case X v = L p ′ ( v ) was shown in [43, Lemma 4.3], which can be used to show (6.34) with γ = 1for the maximal singular integral T .6.3. Muckenhoupt-Wheeden conjecture.
It was conjectured by Muckenhoupt and Whee-den many years ago that for every standard Calder´on-Zygmund operator T , k T f k L , ∞ ( w ) . k f k L ( Mw ) for every weight w. (6.35)A somehow “dual” version of (6.35) can be formulated by (cid:13)(cid:13)(cid:13) T fM w (cid:13)(cid:13)(cid:13) L , ∞ ( w ) . k f k L ( R n ) for every weight w. (6.36)We will present some estimates concerning about (6.36). Recall the operators T and T in(6.28). Thanks to (6.30), Theorem 3.10 applied to X = L , ∞ ( R n ) and the pair ( T f, f ) givesthat (cid:13)(cid:13)(cid:13)(cid:13) T fM w (cid:13)(cid:13)(cid:13)(cid:13) L , ∞ ( Mw ) . (cid:13)(cid:13)(cid:13)(cid:13) fM w (cid:13)(cid:13)(cid:13)(cid:13) L , ∞ ( Mw ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) fM w (cid:13)(cid:13)(cid:13)(cid:13) L ( Mw ) = k f k L ( R n ) . Likewise, treating T , we conclude the following weighted weak-type inequalities. Theorem 6.6.
For every weight w , we have (cid:13)(cid:13)(cid:13)(cid:13) T i fM w (cid:13)(cid:13)(cid:13)(cid:13) L , ∞ ( Mw ) . k M i − f k L ( R n ) , i = 1 , . (6.37) In particular, for every w ∈ A , (cid:13)(cid:13)(cid:13)(cid:13) T i ( f w ) w (cid:13)(cid:13)(cid:13)(cid:13) L , ∞ ( w ) . k M i − f k L ( w ) , i = 1 , . (6.38)The first estimate in (6.37) for the standard Calder´on-Zygmund operator T was proved in[50]. In addition, the first inequality in (6.37) implies that for every weight w , (cid:13)(cid:13)(cid:13)(cid:13) T fM w (cid:13)(cid:13)(cid:13)(cid:13) L , ∞ ( w ) . k f k L ( R n ) , which can be viewed as a substitute of (6.36).On the other hand, we will pay attention to the mixed weak-type estimates: (cid:13)(cid:13)(cid:13)(cid:13) T ( f v ) v (cid:13)(cid:13)(cid:13)(cid:13) L , ∞ ( uv ) . k f k L ( uv ) . (6.39)Such estimates originated in the work of Muckenhoupt and Wheeden [58], where (6.39) for v − ∈ A and uv ≡ M and theHilbert transform H on the real line R . Later, it was developed by Sawyer [66] to the case u, v ∈ A but only for M on R . Also, Sawyer conjectured that (6.39) is true for H and forM but in higher dimensions. A positive answer to both conjectures was given in [20] by usingextrapolation arguments. Recently, Sawyer conjecture for M was extended to the setting of u ∈ A and v ∈ A ∞ in [53] by means of some delicate techniques. Now we formulate Sawyerconjecture for singular integrals on the general Banach function spaces.Now let X uv be a BFS satisfying the hypotheses of Theorem 3.8. Considering (6.30), weutilize Theorem 3.8 to obtain (cid:13)(cid:13)(cid:13)(cid:13) T i ( f v ) v (cid:13)(cid:13)(cid:13)(cid:13) X uv . (cid:13)(cid:13)(cid:13)(cid:13) M i ( f v ) v (cid:13)(cid:13)(cid:13)(cid:13) X uv , i = 1 , . (6.40)Let us go back to the particular case X uv = L , ∞ ( uv ). Let u ∈ A and v ∈ A ∞ . The condition v ∈ A ∞ gives that v ∈ A r for some r >
1, which in turn yields that v = v v − r for some v , v ∈ A . By Lemma 2.1 (7), there exists ε = ε ([ u ] A ) ∈ (0 ,
1) such that uv ε ∈ A forevery ε ∈ (0 , ε ). Then, picking p = 2( r − /ε + 1, we have u − p v = v ( uv r − p − ) − p ∈ A p .By (2.5), M is bounded on L p ( u − p v ), which is equivalent to that M ′ u is bounded on L p ( uv )with a constant C . On the other hand, the condition u ∈ A implies that M ′ u is bounded on L ∞ ( uv ) with the constant C = [ u ] A . Thus, the interpolation theorem [20, Proposition A.1]gives that k M ′ u f k L q, ( uv ) ≤ /q ( C (1 /p − /q ) − + C ) k f k L q, ( uv ) , ∀ q > p . If we set q := 2 p , then k M ′ u f k L q, ( uv ) ≤ p ( C + C ) k f k L q, ( uv ) =: K k f k L q, ( uv ) , ∀ q ≥ q , (6.41)which verifies (3.46). Furthermore, as aforementioned, M ′ v is bounded from L ( uv ) to L , ∞ ( uv ).Hence, this and (6.40) imply the following. WO-WEIGHT EXTRAPOLATION 45
Theorem 6.7.
For every u ∈ A and v ∈ A ∞ with uv ∈ L ( R n ) , (cid:13)(cid:13)(cid:13)(cid:13) T ( f v ) v (cid:13)(cid:13)(cid:13)(cid:13) L , ∞ ( uv ) . k f k L ( uv ) . (6.42)Next, we condiser Sawyer conjecture in the case X uv = L p, ∞ ( uv ) with p > Theorem 6.8.
For every u ∈ A and v ∈ A ∞ with uv ∈ L ( R n ) , there exists ǫ > smallenough such that (cid:13)(cid:13)(cid:13)(cid:13) T i ( f v ) v (cid:13)(cid:13)(cid:13)(cid:13) L ǫ, ∞ ( uv ) . k f k L ǫ, ∞ ( uv ) , i = 1 , . (6.43) Proof.
Let u ∈ A and v ∈ A ∞ . We begin with showing that k M ′ v f k L p, ∞ ( uv ) . p ′ k f k L p, ∞ ( uv ) , ∀ p ∈ (1 , ∞ ) . (6.44)For each N ≥
1, denote u N := u B (0 ,N ) . Then [ u N ] A ≤ [ u ] A for every N ≥
1. For any λ > f λ ( x ) = ( f ( x ) , M ′ v f ( x ) > λ, , M ′ v f ( x ) ≤ λ . We claim that { x ∈ R n : M ′ v f ( x ) > λ } ⊂ { x ∈ R n : M ′ v ( f λ )( x ) > λ } . (6.45)Indeed, fix x ∈ R n such that M ′ v f ( x ) > λ . Then by definition, there exists a cube Q x ∋ x suchthat ffl Q x | f | v dy > λv ( x ). This implies Q x ⊂ { x ∈ R n : M ′ v f ( x ) > λ } , from which we have f λ = f on Q x and hence M ′ v ( f λ )( x ) > λ . Thus, (6.45) holds. Recall that M ′ v is bounded from L ( uv ) to L , ∞ ( uv ) for all u ∈ A and v ∈ A ∞ . Hence, this and (6.45) imply u N v ( { x ∈ R n : M ′ v f ( x ) > λ } ) ≤ u N v ( { x ∈ R n : M ′ v ( f λ )( x ) > λ } ) . λ ˆ R n | f λ | u N v dx = 1 λ ˆ { x ∈ R n : M ′ v f ( x ) >λ } | f | u N v dx. (6.46)Recall that for any 0 < r < p and for any measurable set E with w ( E ) < ∞ , ˆ E | f | r w dx ≤ pp − r w ( E ) − rp k f k rL p, ∞ ( w ) . (6.47)The fact uv ∈ L ( R n ) implies that u N v ( { x ∈ R n : M ′ v f ( x ) > λ } ) ≤ uv ( B (0 , N )) < ∞ . Thus,by (6.46) and (6.47), u N v ( { x ∈ R n : M ′ v f ( x ) > λ } ) . p ′ λ − k f k L p, ∞ ( u N v ) u N v ( { x ∈ R n : M ′ v f ( x ) > λ } ) − p , and so we get λ u N v ( { x ∈ R n : M ′ v f ( x ) > λ } ) p . p ′ k f k L p, ∞ ( uv ) . Accordingly, it follows from the monotone convergence theorem that (6.44) holds. Applying(3.49), (6.30) and (6.44), we deduce that (cid:13)(cid:13)(cid:13)(cid:13) T i ( f v ) v (cid:13)(cid:13)(cid:13)(cid:13) L ǫ, ∞ ( uv ) . (cid:13)(cid:13)(cid:13)(cid:13) M i ( f v ) v (cid:13)(cid:13)(cid:13)(cid:13) L ǫ, ∞ ( uv ) = (cid:13)(cid:13) ( M ′ v ) i f (cid:13)(cid:13) L ǫ, ∞ ( uv ) . k f k L ǫ, ∞ ( uv ) . This shows (6.43) and completes the proof. (cid:3)
Littlewood-Paley operators.
Given ρ ∈ C , Ω ∈ L ( S n − ), a radial function h on R n ,and λ >
1, we define the parametric Marcinkiewicz integrals µ ρ Ω ,h,S ( f )( x ) := (cid:18) ¨ Γ( x ) | Θ ρ Ω ,h f ( y, t ) | dydtt n +1 (cid:19) ,µ ∗ ,ρ Ω ,h,λ ( f )( x ) := (cid:18) ¨ R n +1+ (cid:16) tt + | x − y | (cid:17) nλ | Θ ρ Ω ,h f ( y, t ) | dydtt n +1 (cid:19) , where Γ( x ) := { ( y, t ) ∈ R n +1+ : | x − y | < t } andΘ ρ Ω ,h f ( x, t ) = 1 t ρ ˆ | x − y |≤ t Ω( x − y ) h ( x − y ) | x − y | n − ρ f ( y ) dy. If h ≡
1, the operator µ ρ Ω ,h,S was introduced by H¨ormander [38] in the higher dimension.If ρ = 1 and h ≡
1, it is the usual Marcinkiewicz integral corresponding to the Littlewood-Paley g -function introduced by Stein [68], where L p -boundedness (1 < p ≤
2) of µ , ,S wasestablished for Ω ∈ Lip α ( S n − ) (0 < α := Re( ρ ) ≤ ρ > < p < ∞ in [38].For 1 ≤ q ≤ ∞ , denote ℓ ∞ ( L q )( R + ) := (cid:26) h ∈ L ( R + ) : sup j ∈ Z (cid:18) ˆ j +1 j | h ( r ) | drr (cid:19) q < ∞ (cid:27) . When q = ∞ , ℓ ∞ ( L q )( R + ) is understood as L ∞ ( R + ).In [26, Lemma 3], Ding et al. showed that if λ > ρ ∈ C with Re( ρ ) > h ∈ ℓ ∞ ( L q )( R + )with q ∈ (1 , ∞ ] and Ω ∈ L log + L ( S n − ) is homogeneous of degree zero on R n and ´ S n − Ω dσ =0, then for every weight w , k µ ∗ ,ρ Ω ,h,λ ( f ) k L ( w ) ≤ C n,ρ,λ k f k L ( Mw ) . (6.48)Observe that for any λ > µ ρ Ω ,h,S ( f )( x ) ≤ nλ µ ∗ ,ρ Ω ,h,λ ( f )( x ) , x ∈ R n . (6.49)In view of Theorem 3.11, (6.48) and (6.49) give the following conclusion. Theorem 6.9.
Let λ > , ρ ∈ C with Re( ρ ) > , a radial function h ∈ ℓ ∞ ( L q )( R + ) with q ∈ (1 , ∞ ] . Let Ω ∈ L log + L ( S n − ) be homogeneous of degree zero on R n and ´ S n − Ω dσ = 0 .Suppose that X be a RIBFS over ( R n , dx ) with q X < ∞ such that X is also a BFS. Then forevery weight w , k µ ρ Ω ,h,S ( f ) k X w ≤ C k f ( M w/w ) k X w , (6.50) k µ ∗ ,ρ Ω ,h,λ ( f ) k X w ≤ C k f ( M w/w ) k X w . (6.51) Remark 6.10.
Theorem . covers several known results in Lebesgue space. Let X = L p ( R n ) and ≤ p < ∞ . Then we see that both X and X are BFS and q X = p . Hence, Theorem . contains the unweighted inequalities in Theorems and and the weighted L inequalityfor A weight in Corollary 1 in [26] . Moreover, (6.50) and (6.51) extend the Fefferman-Steininequalities in Theorem and Corollary in [71] to the general Banach function spaces. WO-WEIGHT EXTRAPOLATION 47
Now let us turn to another type of square functions. Given α, β ∈ R and λ >
0, we definethe square functions by g α,β ( f )( x ) := (cid:18) ¨ Γ α ( x ) | f ∗ φ t ( y ) | dyt (1 − α ) n +2 β dtt (cid:19) ,g ∗ α,β,λ ( f )( x ) := (cid:18) ˆ t α ≤ ˆ R n (cid:16) t − α t − α + | x − y | (cid:17) nλ | f ∗ φ t ( y ) | dyt (1 − α ) n +2 β dtt (cid:19) , where φ t ( x ) = t n φ ( xt ), φ is a smooth function with suitable compact Fourier support awayfrom the origin, and Γ α ( x ) := { ( y, t ) ∈ R n × R + : 0 < t α ≤ , | y − x | ≤ t − α } . A new type offractional maximal operator is given by M α,β f ( x ) := sup ( y,r ) ∈ Γ α ( x ) | B ( y, r ) | − β/n ˆ B ( y,r ) | f ( z ) | dz. (6.52)Observe that g ∗ α,β,λ is a pointwise majorant of g α,β . If α = β = 0, then M α,β is the Hardy-Littlewood maximal function, g α,β and g ∗ α,β,λ are the generalization of Littlewood-Paley oper-ators with Poisson kernels studied in [69].Recently, Beltran and Bennett [6] proved that for every α, β ∈ R and every weight w , ˆ R n | f ( x ) | w ( x ) dx ≤ C ˆ R n g α,β f ( x ) M α,β M w ( x ) dx, (6.53)for all functions f such that supp( b f ) ⊂ { ξ ∈ R n : | ξ | α ≥ } . Conversely, they obtained thatfor all α ∈ R , λ >
1, and for every weight w , ˆ R n g ∗ α, ,λ f ( x ) w ( x ) dx ≤ C ˆ R n | f ( x ) | M w ( x ) dx. (6.54)By Theorem 3.12, (3.91) and (6.54), we get the two-weight inequalities on the weightedBanach function spaces as follows. Theorem 6.11.
Let u , v , w and w be weights on R n . Suppose that X u and X v are respectivelyBFS over ( R n , u dx ) and ( R n , v dx ) such that Y u = X u and Y v = X v are BFS. Assume that k ( M L log L f ) w − v − k Y ′ v ≤ C k f w − u − k Y ′ u , ∀ f ∈ M . (6.55) Then for all α ∈ R and λ > , k ( g ∗ α, ,λ f ) w k X u ≤ C k f w k X v . (6.56)6.5. Fourier integral operators.
For a function m on R n , the Fourier multiplier T m is definedby T m f ( x ) := ˆ R n m ( ξ ) b f ( ξ ) e − πix · ξ dξ, for all functions f ∈ S ( R n ). For each α, β ∈ R , let C ( α, β ) be the class of functions m : R → C for which supp( m ) ⊂ { ξ : | ξ | α ≥ } , sup ξ ∈ R n | ξ | β | m ( ξ ) | < ∞ , andsup r α ≥ sup I ⊂ [ r, r ] ℓ ( I )= r − α r β ˆ ± I | m ′ ( ξ ) | dξ < ∞ . Let D ( α, β ) be the collection of all functions m : R n → C such thatsup B dist(0 , B ) β +(1 − α ) θ | B | − k m Ψ B k ˙ H θ < ∞ , for all 0 ≤ θ ≤ σ and some σ > n/
2, uniformly over normalized bump functions Ψ B adaptedto an α -subdyadic ball B . Here, ˙ H θ denotes the usual homogeneous Sobolev spaces of order θ , and Ψ is a suitable smooth function with compact support away from the origin. By the α -subdyadic ball, we mean that a Euclidean ball B ⊂ R n satisfies that dist(0 , B ) α ≥ r ( B ) ≃ dist(0 , B ) − α .Bennett [7] showed that for every m ∈ C ( α, β ) with α, β ∈ R and for every weight w , ˆ R | T m f | w dx ≤ C ˆ R | f | M M α,β M w dx, (6.57)where M α,β is defined in (6.52). By H¨older’s inequality, one has M α,β w ( x ) ≤ sup x ∈ B ( y,r − α ) r β (cid:18) B ( y,r ) w s dz (cid:19) s ≤ sup x ∈ B ( y,r − α ) r β − αs (cid:18) B ( y,r − α ) w s dz (cid:19) s ≤ M ( w s )( x ) s , (6.58)provided α = 2 sβ and s ≥
1. Therefore, from (6.57) and (6.58), we obtain that for every m ∈ C ( α, α/
2) and for every weight w ˆ R | T m f | w dx ≤ C ˆ R | f | M w dx. (6.59)In addition, it was shown in [6, Theorem 1] that for every m ∈ D ( α, β ) with α, β ∈ R , g α,β ( T m f )( x ) . g ∗ α, ,λ ( f )( x ) , λ = 2 σ/n > . (6.60)Then invoking (6.53), (6.60), (6.54) and (6.58), we conclude that for every weight w , ˆ R n | T m f | w dx ≤ C ˆ R n | f | M M α,α/ M w dx ≤ C ˆ R n | f | M w dx, (6.61)for any m ∈ D ( α, α/
2) supported in { ξ ∈ R n : | ξ | α ≥ } .Recall the pseudo-differential operator T σ defined in (6.3). It was proved in [5] that if σ ∈ S m̺,δ with m ∈ R , 0 ≤ δ ≤ ̺ ≤ δ <
1, then for any weight w , ˆ R n | T σ f | w dx . ˆ R n | f | M M ̺,m M w dx, (6.62)where M ̺,m w ( x ) := sup ( y,r ) ∈ Λ ̺ ( x ) w ( B ( y, r )) | B ( y, r ) | m/n , and Λ ̺ ( x ) := { ( y, r ) ∈ R n × (0 ,
1) : | y − x | ≤ r ̺ } . Observe that B ( y, r ) ⊂ B ( x, r ̺ ) for any( y, r ) ∈ Λ ̺ ( x ). Thus, picking m ∈ R and ̺ ∈ [0 ,
1] such that ̺ = 1 + 2 m/n , we get M ̺,m w ( x ) ≤ sup ( y,r ) ∈ Λ ̺ ( x ) | B ( x, r ̺ ) || B ( y, r ) | m/n w ( B ( x, r ̺ )) | B ( x, r ̺ ) | ≤ CM w ( x ) . (6.63)Gathering (6.62) and (6.63), we conclude that ˆ R n | T σ f | w dx ≤ C ˆ R n | f | M w dx. (6.64) WO-WEIGHT EXTRAPOLATION 49
As a consequence, combining (6.59), (6.61), (6.64) and Theorem 3.14, we conclude the followingestimates.
Theorem 6.12.
Let v be a weight on R n , X v be a BFS over ( R n , v dx ) such that X v is alsoa BFS. Assume that there exist Young functions A and B such that A − ( t ) B − ( t ) . Φ − ( t ) ,and that M ′ B,v is bounded on ( X v ) ′ . Then for every weight u , k ( T f ) u k X v ≤ C k f M A ( u ) k X v , (6.65) provided that the pair ( T , Φ) satisfies one of the following: (1) ( T , Φ) = ( T m , t log( e + t ) ) , where m ∈ C ( α, α/ with α ∈ R ; (2) ( T , Φ) = ( T m , t log( e + t ) ) , where m ∈ D ( α, α/ supported in { ξ : | ξ | α ≥ } with α ∈ R ; (3) ( T , Φ) = ( T σ , t log( e + t ) ) , where σ ∈ S m̺,δ with m = − n (1 − ̺ ) / , ≤ δ ≤ ̺ ≤ and δ < . References [1] N. Accomazzo, J.C. Mart´ınez-Perales, I.P. Rivera-R´ıos,
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Email address : [email protected] Andrea Olivo, Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Uni-versidad de Buenos Aires and IMAS-CONICET, Pabell´on I (C1428EGA), Ciudad de Buenos Aires,Argentina
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