Weighted estimates for the multilinear maximal function

arXiv:1304.5999v2 [math.CA] 9 Jul 2013 WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION WEI CHEN AND WENDOL´ IN DAMI´ AN Abstract. A formulation of the Carleson embedding theorem in the multilinear setting is proved which allows to obtain a multilinear analogue of Sawyer’s two weight theorem for the multisublinear maximal function Mintroduced in [8]. A multilinear version of the Bptheorem from [6] is also obtained and a mixed A~ P−W∞ ~ Pbound for Mis proved as well. 1. Introduction The beginning of the modern theory of weights was originated in the works of R. Hunt, B. Muckenhoupt, R. Wheeden, R. Coifman and C. Fefferman in the decade of the 70’s. In [12] B. Muckenhoupt characterized the class of weights u, v for which the following weak inequality holds (1.1) sup λ>0 λpZ{Mf>λ} u(x)dx ≤CZRn |f(x)|pv(x)dx, f ∈Lp(v), where Mdenotes the Hardy–Littlewood maximal operator and p≥1. This condition on the weights is known as Apcondition, namely [u, v]Ap:= sup Q1 |Q|ZQ u(x)dx 1 |Q|ZQ v(x)−1 p−1p−1 <∞, p > 1 where the supremum is taken over all the cubes in Rn. When p= 1, the term (RQ 1 |Q|v(x)−1 p−1)p−1must be understood as (infQv)−1. In the particular case when u=v, Muckenhoupt also proved that the following strong estimate ZRn (Mf(x))pv(x)dx ≤CZRn |f(x)|pv(x)dx, f ∈Lp(v), The first author is supported by the National Natural Science Foundation of China (Grant No. 11101353), the Natural Science Foundation of Jiangsu Education Committee (Grant No. 11KJB110018) and the Natural Science Foundation of Jiangsu Province (Grant No. BK2012682). The second author is supported by Junta de Andaluc´ıa (Grant No. P09-FQM-4745) 2010 Mathematics Subject Classification. 42B25. Key words and phrases. Multilinear maximal operator, weighted bounds, reverse H¨older’s inequality. 1 2 W. CHEN AND W. DAMI ´ AN holds if and only if vsatisfies the Apcondition. However, the problem of finding a condition on the weights u, v satisfying the strong estimate above was more complicated. In [13] E. Sawyer characterized the two weight inequality, showing that M:Lp(v)−→ Lp(u) if and only if the pair (u, v) satisfies the following testing condition known as Sawyer’s Spcondition (1.2) [u, v]Sp= sup Q RQM(χQσ)pudx σ(Q)!1/p <∞, where σ=v1−p′and 1 < p < ∞. Motivated by these results the theory of weighted inequalities developed rapidly, not only for the Hardy–Littlewood maximal operator but also for some of the main operators in Harmonic Analysis like Calder´on– Zygmund operators. Much later the interest focused in determining the sharp dependence of the Lp(w) operator norm in term of the relevant constant involving the weights. On this point, the problem for the Hardy–Littlewood maximal operator was solved by S. Buckley [1] who proved (1.3) ||M||Lp(w)≤C p′[w] 1 p−1 Ap, where Cis a dimensional constant. Motivated by this result and others, K. Moen found in [11] a quantitative form of E. Sawyer’s result mentioned above in terms of Sawyer’s condition (1.2), namely (1.4) ||M||Lp(v)−→Lp(u)≈[u, v]Sp. Recently, T. Hyt¨onen and C. P´erez in [6] (see also [7] for a better result and a simplified proof) improved Buckley’s bound (1.3) replacing a portion of the Ap constant by the weaker A∞constant as defined by Fujii in [4] and later used in the work of J.M. Wilson [14]. The A∞constant is defined as follows (1.5) [w]A∞:= sup Q 1 w(Q)ZQ M(wχQ), where the supremum is taken over all the cubes Qin Rn. In [6] the authors show in a two-weight setting and for p > 1 that (1.6) ||M(fσ)||Lp(w)≤Cp′(Bp[w, σ])1/p||f||Lp(σ), and (1.7) ||M(fσ)||Lp(w)≤Cp′([w]Ap[σ]A∞)1/p||f||Lp(σ), where Cin both inequalities is a dimensional constant and WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION 3 (1.8) Bp[w, σ] := sup Q1 |Q|ZQ w 1 |Q|ZQ σp exp 1 |Q|ZQ log σ−1, is known as the Bpconstant of the weights wand σ. This constant clearly satisfies [w]Ap≤Bp[w, σ]≤[w]Ap[σ]′ A∞, where [σ]′ A∞denotes the A∞constant introduced by S. Hrusˇcˇev in [5] defined as follows [w]′ A∞= sup Q1 |Q|ZQ wexp 1 |Q|ZQ log w−1. In the one weight setting and by a standard change-of-weight argument, (1.7) implies (1.9) ||M||Lp(w)≤Cp′([w]Ap[σ]A∞)1/p, where σ=w1−p′. The aim of this article is to give some multilinear analogues of some of the above mentioned results following the spirit of the theory of multiple weights developed in [8]. The paper is organized as follows. Some preliminary definitions and results are summarized in Sect. 2. In Sect. 3we give the statements of the main results on this paper and some remarks on them. Finally, in Sect. 4we give all the proofs of our results. Throughout this paper, we will use the notation A.Bto indicate that there is a constant c, independent of the weight constant, such that A≤cB. 2. Preliminaries Before stating and proving our main results, we first recall some basics related to the theory of multilinear weighted inequalities as well as we introduce the definition of some constants involved in the multiple theory of weights and dyadic grids. 2.1. Some basics on multilinear weighted inequalities. One of the main objects of the theory of multiple weights is the following extension of the classical Hardy–Littlewood maximal function. Given −→ f= (f1,...,fm), we define following [8] the multi(sub)linear maximal operator Mby M(−→ f)(x) = sup Q∋x m Y i=1 1 |Q|ZQ |fi(yi)|dyi, where the supremum is taken over all cubes Qcontaining x. The importance of this operator stems from the fact that it controls the class of multilinear Calder´on– Zygmund operators as it is shown in [8]. A particular example of this relationship is the class of weights characterizing the weighted Lpspaces for which both operators 4 W. CHEN AND W. DAMI ´ AN are bounded. To define this class of weights we let −→ w= (w1,...,wm) and −→ P= (p1,...,pm) such that 1 < p1,...,pm<∞. Set 1 p=1 p1+···+1 pmand ν−→ w= Qm i=1 wp/pi i. We say that −→ wsatisfies the A−→ Pcondition if [−→ w]A−→ P= sup Q1 |Q|ZQ ν−→ wm Y i=1 1 |Q|ZQ w1−p′ i ip/p′ i<∞. It is easy to see that in the linear case (that is, if m= 1) [−→ w]A−→ P= [w]Apis the usual Apconstant. In [8] the following multilinear extension of the Muckenhoupt Aptheorem for the maximal function was obtained: the inequality (2.1) kM(−→ f)kLp(ν−→ w)≤C m Y i=1 kfikLpi(wi) holds for every −→ fif and only if −→ wsatisfies the A−→ Pcondition. Very recently in [3] A. Lerner, C. P´erez and the second author proved a multilinear version of Buckley’s result as well as a full analogue of (1.7). In this work the authors found that the multilinear version of (1.7) is sharp when m≥1, although is much more complicated to do the same for Buckley’s result. In this case, several partial results were obtained in [3] which have been improved in [9]. 2.2. Some constants on multiple weight theory. Next we state the notation that we will follow in the sequel related to some constants involved in the multiple theory of weights. To define these constants, let w1,...,wmand vbe weights and let us denote −→ w= (w1,...,wm). Also let 1 < p1,...,pm<∞and pbe numbers such that 1 p=1 p1+···+1 pmand denote −→ P= (p1,...,pm). We say that (v, −→ w) satisfies the A−→ Pcondition if (2.2) [v, −→ w]A−→ P:= sup Q1 |Q|ZQ vm Y i=1 1 |Q|ZQ w1−p′ i ip/p′ i<∞. In particular when v=ν−→ w:= Qm i=1 w p pi i, we will write [ν−→ w,−→ w]A−→ Pas [−→ w]A−→ P. Next we define the multilinear analogues of the A∞constant defined by Fujii in [4], the Bpconstant defined by Hyt¨onen and P´erez in [6] and the Spconstant defined by Sawyer in [13], respectively. We say that (1) −→ wsatisfies the W∞ −→ Pcondition if [−→ w]W∞ −→ P= sup QZQ m Y i=1 M(wiχQ)p pidxZQ m Y i=1 w p pi idx−1 <∞. WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION 5 (2) (v, −→ w) satisfies the B−→ Pcondition if [v, −→ w]B−→ P:= sup Q v(Q) |Q|m Y i=1 wi(Q) |Q|pexp 1 |Q|ZQ log m Y i=1 w−p pi idx<∞. (3) (v, −→ w) satisfies the S−→ Pcondition if [v, −→ w]S−→ P= sup QZQ M(−−→ σχQ)pvdx1 pm Y i=1 σi(Q)1 pi−1 <∞, where −−→ σχQ= (σ1χQ,...,σmχQ) and σi=w1−p′ i ifor all i= 1,...,m and all the suprema in the above definitions are taken over all cubes Qin Rn. Additionally we define a multiple Reverse H¨older condition that we will use in the following. We say that −→ wsatisfies the RH−→ Pcondition if there exists a positive constant Csuch that (2.3) m Y i=1 ZQ σidxp pi≤CZQ m Y i=1 σ p pi idx, where σi=w1−p′ i ifor i= 1,...,m. We denote by [−→ w]RH−→ Pthe smallest constant C in (2.3). 2.3. Dyadic grids. Recall that the standard dyadic grid Din Rnconsists of the cubes 2−k([0,1)n+j), k ∈Z, j ∈Zn. By a general dyadic grid Dwe mean a collection of cubes with the following properties: (1) For any Q∈Dits sidelength ℓQis 2k, k ∈Z (2) Q∩R∈ {Q, R, ∅} for any Q, R ∈D. (3) The cubes of a fixed sidelength 2kform a partition of Rn. We say that {Qk j}is a sparse family of cubes if: (1) The cubes Qk jare disjoint in j, with kfixed. (2) If Ωk=∪jQk j, then Ωk+1 ⊂Ωk. (3) |Ωk+1 ∩Qk j| ≤ 1 2|Qk j|. With each sparse family {Qk j}we associate the sets Ek j=Qk j\Ωk+1. Observe that the sets Ek jare pairwise disjoint and |Qk j| ≤ 2|Ek j|. In the sequel we will use the following lemmas that could be found in [6] and [3], respectively. Lemma 2.1. There are 2ndyadic grids Dαsuch that for any cube Q⊂Rnthere exists a cube Qα∈Dαsuch that Q⊂Qαand ℓQα≤6ℓQ. 6 W. CHEN AND W. DAMI ´ AN Lemma 2.2. For any non-negative integrable fi, i = 1,...,m, there exist sparse families Sα∈Dαsuch that for all x∈Rn, M(−→ f)(x)≤(2 ·12n)m 2n X α=1 ADα,Sα(−→ f)(x), where −→ f= (f1,...,fm)and given a sparse family S={Qk j}of cubes from a dyadic grid D, the operator AD,Sis given by AD,S(−→ f) = X j,k m Y i=1 (fi)Qk j!χQk j. 3. Main results In this section we summarize the main results on this work. Firstly we state the main tool of this paper. This lemma extends to the multilinear setting a nonstandard formulation of the (dyadic) Carleson embedding theorem proved in [6] and it will allow us to prove our main results. Lemma 3.1. Suppose that the nonnegative numbers {aQ}Qsatisfy (3.1) X Q⊂R aQ≤AZR m Y i=1 σ p pi idx, ∀R∈D where σiare weights for i= 1,...,m. Then for all 1< pi<∞and p∈(1,∞) satisfying 1 p=1 p1+···+1 pmand for all fi∈Lpi(σi), X Q∈D aQm Y i=1 1 σi(Q)ZQ fi(yi)σi(yi)dyip!1/p ≤A||Md −→ σ(−→ f)||Lp(ν−→ σ) ≤A m Y i=1 p′ i||fi||Lpi(σi), (3.2) where Md −→ σ(−→ f) = sup Q∋x Q∈D m Y i=1 1 σi(Q)ZQ |fi(yi)|σi(yi)dyi. Next we establish a generalization of Sawyer’s theorem to the multilinear setting. Very recently it was shown in [10] a multilinear version of Sawyer’s theorem using a kind of monotone property on the weights. We establish here another condition that is a sort of reverse H¨older inequality in the multilinear setting (see Sect. 2for definition) and that was used by the first author in [2] in the setting of martingale spaces. When m= 1 this reverse H¨older condition is superfluous and we recover the linear result of Moen (1.4). WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION 7 Theorem 3.2. Let 1< pi<∞,i= 1,...,m and 1 p=1 p1+...+1 pm. Let vand wibe weights. If we suppose that −→ w∈RH−→ Pthen there exists a positive constant C such that (3.3) ||M(−→ fσ)||Lp(v)≤C m Y i=1 ||fi||Lpi(σi), fi∈Lpi(σi), where σi=w1−p′ i i, if and only if (v, −→ w)∈S−→ P. Moreover, if we denote the smallest constant Cin (3.3)by ||M||, we obtain (3.4) [v, −→ w]S−→ P.||M|| .[v, −→ w]S−→ P[−→ w]1/p RH−→ P . Here we make some remarks related to the previous theorem. Remark 3.3. In the particular case when v=ν−→ w, the following statements are equivalent: (1) −→ w∈A−→ P. (2) σi=w1−p′ i i∈Amp′ i, for i= 1,...,m and ν−→ w∈Amp. (3) (ν−→ w,−→ w)∈S−→ P. (4) There exists a positive constant Csuch that (3.5) ||M(−→ f)||Lp(ν−→ w)≤C m Y i=1 ||fi||Lpi(wi), fi∈Lpi(wi). Indeed, the equivalence between 1., 2.and 4.was proved in [8, Th. 3.6, Th. 3.7]. It can be easily seen that in this particular case [ν−→ w,−→ w]S−→ P.||M|| where ||M|| denotes the smallest constant in (3.5) and [−→ w]A−→ P.[ν−→ w,−→ w]p S−→ P. Therefore we have that 4.implies 3.and 3.implies 1.. So we have obtained that all the statements are equivalent. Additionally, following [3, Th. 1.1], we also have that ||M|| .[−→ w]1/p A−→ PQm i=1[σi] 1 pi ∞. So, we have obtained (3.6) [−→ w]1/p A−→ P.[v−→ w,−→ w]S−→ P.||M|| .[−→ w]1/p A−→ P m Y i=1 [σi] 1 pi ∞. Remark 3.4. As we have observed in the previous remark, RH−→ Pcondition is not necessary when v=ν−→ win Theorem 3.2. We are not sure if this condition can be removed in the general case. Making use of the analogue of the Bpconstant within the multilinear setting already defined in Sect. 2, we obtain an extension of (1.6). 8 W. CHEN AND W. DAMI ´ AN Theorem 3.5. Let 1< pi<∞,i= 1,...,m and 1 p=1 p1+...+1 pm. Let vand wi be weights. Then (3.7) ||M(−→ fσ)||Lp(v).[v, −→ σ]1/p B−→ P m Y i=1 ||fi||Lpi(σi), fi∈Lpi(σi), where σi=w1−p′ i i,−→ σ= (σ1,...,σm)and −→ fσ = (f1σ1,...,fmσm). And finally, using the generalization of the Fujii–Wilson A∞constant [−→ w]W∞ −→ Pand the two-weight constant [v, −→ w]A−→ Pdefined in Sect. 2, we get a mixed A−→ P−W∞ −→ P bound for Mthat extends (1.7) to the multilinear setting. Theorem 3.6. Let 1< pi<∞,i= 1,...,m and 1 p=1 p1+...+1 pm. Let vand wi be weights. Then (3.8) ||M(−→ fσ)||Lp(v).([v, −→ w]A−→ P[−→ σ]W∞ −→ P)1/p m Y i=1 ||fi||Lpi(σi), fi∈Lpi(σi), where σi=w1−p′ i i,−→ σ= (σ1,...,σm)and −→ fσ = (f1σ1,...,fmσm). 4. Proofs We start proving Lemma 3.1, that is, the multilinear version of the dyadic Carleson embedding theorem. It follows a scheme of proof similar to the one used by Hyt¨onen and P´erez in [6]. Lemma 3.1.Let us see the sum X Q∈D aQ m Y i=1 1 σi(Q)ZQ fi(yi)σi(yi)dyi!p as an integral on a measure space (D,2D, µ) built over the set of dyadic cubes D, assigning to each Q∈Dthe measure aQ. Thus X Q∈D aQ m Y i=1 1 σi(Q)ZQ fi(yi)σi(yi)dyi!p = =Z∞ 0 pλp−1µ(Q∈D: m Y i=1 1 σi(Q)ZQ fi(yi)σi(yi)dyi> λ) =: Z∞ 0 pλp−1µ(Dλ)dλ. WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION 9 Let us denote by D∗ λthe set of maximal dyadic cubes Rwith the property that Qm i=1 1 σi(Q)RRfi(yi)σi(yi)dyi> λ. Then the cubes R∈D∗ λare disjoint and their union is equal to the set {Md −→ σ(−→ f)> λ}. Thus µ(Dλ) = X Q∈Dλ aQ≤X R∈D∗ λX Q⊂R aQ ≤AX R∈D∗ λZR m Y i=1 σ p pi idx =AZ{Md −→ σ(−→ f)>λ} m Y i=1 σ p pi idx. Then we obtain X Q∈D aQ m Y i=1 1 σi(Q)ZQ fi(yi)σi(yi)dyi!p ≤AZ∞ 0 pλp−1Z{Md −→ σ(−→ f)>λ} m Y i=1 σ p pi idxdλ =AZRn Md −→ σ(−→ f)p m Y i=1 σ p pi idx ≤AZRn m Y i=1 ((Md σi(fi))piσi)p pidx ≤A m Y i=1 ZRn (Md σi(fi))piσidxp pi ≤A m Y i=1 (p′ i)pZRn |fi|piσidxp pi, where we have used that Md −→ σ(−→ f)≤Qm i=1 Md σi(fi), H¨older’s inequality and the boundedness properties of Md σi(fi) in Lpi(σi).  Next we prove Theorem 3.2 making use of Lemma 3.1. Theorem 3.2.It is clear that (3.3) implies the S−→ Pcondition without using that (v, −→ w)∈RH−→ P. Thus, it remains to prove that (v, −→ w)∈S−→ Pimplies (3.3) to complete the proof of the theorem. By Lemma 2.1, it suffices to prove the theorem for the dyadic maximal operators MDα. Since the proof is independent of the particular dyadic grid, without loss of generality we consider Mdtaken with respect to the standard dyadic grid D. Next we proceed as in the proof of Lemma 2.2. Let a= 2m(n+1) and for k∈Zconsider the following sets