The endpoint Fefferman-Stein inequality for the strong maximal function

arXiv:1211.2950v2 [math.CA] 1 Oct 2013 THE ENDPOINT FEFFERMAN-STEIN INEQUALITY FOR THE STRONG MAXIMAL FUNCTION TERESA LUQUE AND IOANNIS PARISSIS Abstract. Let Mnfdenote the strong maximal function of fon Rn, that is the maximal average of fwith respect to n-dimensional rectangles with sides parallel to the coordinate axes. For any dimension n⩾2 we prove the natural endpoint Fefferman-Stein inequality for Mnand any strong Muckenhoupt weight w: w({x∈Rn:Mnf(x)> λ}).w,nZRn |f(x)| λ1+log+|f(x)| λn−1Mnw(x)dx. This extends the corresponding two-dimensional result of T. Mitsis. 1. Introduction The strong maximal function. Let Rndenote the family of all rectangles in Rnwith sides parallel to the coordinate axes. For a locally integrable function fon Rnwe will denote by Mnf the strong maximal function: Mnf(x)≔sup R∈Rn R∋x 1 |R|ZR |f(y)|dy,x∈Rn. Here |S|denotes the n-dimensional Lebesgue measure of a set S⊂Rn. We will sometimes use the same notation for the (n−1)-dimensional Lebesgue measure, but this will be clear from the context. The endpoint behavior of Mnclose to L1is given by the classical theorem of Marcinkiewicz, Jessen and Zygmund, [8]: |{x∈Rn:Mnf(x)> λ}| .nZRn |f(x)| λ1+log+|f(x)| λn−1dx,(1.1) where log+t≔max(0, log t). Inequality (1.1) implies the strong differentiation of the integral of all functions f∈L(1+ (log+L)n−1)(Rn), that is, all functions fon Rnsuch that: ZRn |f(x)|1+log+|f(x)|n−1dx < +∞. The strong maximal theorem cannot be improved, that is, the function t(1+ (log+t)n−1)on the right hand side of (1.1) cannot be replaced by any slower increasing function. Remember 2010 Mathematics Subject Classification. Primary: 42B25, Secondary: 42B20. T.L. is supported by the Spanish Ministry of Economy and Competitiveness grant BES-2010-030264. I.P. is supported by the Academy of Finland, grant 138738. 1 2 T. LUQUE AND I. PARISSIS that the usual Hardy-Littlewood maximal function is the maximal average of fwith respect to all n-dimensional Euclidean cubes, or balls and it maps L1(Rn)to L1,∞(Rn). The important difference to be noted here is that the strong maximal function is an n-parameter maximal average, in contrast to the the usual one-parameter Hardy-Littlewood maximal function, and this difference is reflected in the strong maximal theorem (1.1) which requires extra logarithmic scales of integrability. The original proof from [8] relies on the observation that Mncan be viewed as a composition of none-dimensional Hardy-Littlewood maximal operators. One then appeals to the one-dimensional theory to get (1.1) together with its strong Lp(Rn)counterparts. A more geometric point of view was introduced by the work of Córdoba and R. Fefferman, [3], who gave a proof of (1.1) by means of a geometric covering argument. This is in a sense a dual point of view where the n-parameter composition of operators is replaced by induction on the dimension. The importance of the Córdoba-Fefferman geometric proof of the strong maximal theorem is highlighted by the fact that the usual Besicovitch covering argument fails when applied to families of rectangles having arbitrary eccentricities. Strong weights. Aweight wwill be a locally integrable, non-negative function on Rn. We will say that wbelongs to the class A∗ p, 1 < p < ∞, whenever [w]A∗ p≔sup R∈Rn1 |R|ZR w 1 |R|ZR w1−p′p−1 <+∞. In this case we will say that wis a strong Ap-weight. For p=1 the class A∗ 1is defined by the condition 1 |R|ZR w⩽Cess inf x∈Rw(x), for almost every x∈R,R∈Rn, which is equivalent to saying that Mnw⩽Cw almost everywhere in Rn. The smallest constant C > 0 in the previous inequality is the A∗ 1-constant of the weight, denoted by [w]A∗ 1. It follows by Hölder’s inequality that the A∗ pclasses are increasing, that is, for 1 ⩽p⩽q < ∞ we have A∗ p⊂A∗ q. We define the class A∗ ∞by means of A∗ ∞≔[ p>1 A∗ p. It is equivalent to define the class A∗ ∞by the following property: there exist constants δ,c > 0 such that, given any rectangle R∈Rnand a measurable subset S⊂R, then w(S) w(R)⩽c|S| |R|δ .(1.2) An important feature of strong A∞-weights is that if we fix any t∈Rthen the weight wt(x′)≔w(x′,t),x′∈Rn−1, is an A∗ ∞-weight on Rn−1,uniformly in t∈R. In practice, uniformly means that all the constants connected with the properties of the A∗ ∞-weight wtcan be taken to be independent of t. For these and other properties of strong Muckenhoupt weights see for example [2] or [6, Chapter IV]. THE ENDPOINT FEFFERMAN-STEIN INEQUALITY FOR THE STRONG MAXIMAL FUNCTION 3 Remark 1.3.Let w∈A∗ ∞. By the previous discussion we see that there exists some ǫ=ǫ(w)>0 such that, for every rectangle R∈Rnand all measurable sets F⊂Rn, we have |R∩F|⩽ǫ|R|⇒w(R∩F)⩽1 2w(R)⇒w(R\F)⩾1 2w(R). In fact, it suffices to choose ǫ > 0 so that cǫδ⩽1 2, where c,δare the constants associated to w∈A∗ ∞from (1.2). Since for any t∈Rthe weight wt≔w(·,t)is an A∗ ∞-weight on Rn−1, uniformly in t, the ǫ > 0 can be chosen sufficiently small so that we have the previous property also for wt, rectangles R′∈Rn−1and sets F′⊂Rn−1, uniformly in t. We will use this remark several times in what follows. It is known that Mnis bounded on Lp(w), 1 < p < ∞if and only if w∈A∗ p. This result follows again by an appeal to the one dimensional theory. For the necessity of the A∗ pcondition, one argues as in the case of the usual Apweights. The corresponding endpoint bound is also true: the operator Mnsatisfies the distributional estimate w({x∈Rn:Mnf(x)> λ}).w,nZRn |f(x)| λ1+log+|f(x)| λn−1w(x)dx whenever w∈A∗ 1. For these results see [2, Theorems 2.1 and 2.3]. Weighted strong maximal function. For w∈A∗ ∞we will also consider the weighted strong maximal function Mw n, defined with respect to w: Mw nf(x)≔sup R∈Rn R∋x 1 w(R)ZR |f(y)|w(y)dy. For the weighted strong maximal function Mw n, R. Fefferman showed in [5] that it maps Lp(w) to Lp(w)whenever w∈A∗ ∞: kMw nfkLp(w)⩽w,ncp,nkfkLp(w), 1 < p ⩽∞.(1.4) Furthermore we have the asymptotic estimate cp,n=On((p−1)−n)as p→1+.(1.5) The behavior of the constants is not explicitly studied in [5] but follows by a close examination of the proof and the standard norm bounds from Marcinkiewicz interpolation. See also [10, Lemma 4]. The endpoint bound for Mw nis also true, namely Mw nsatisfies w({x∈Rn:Mw nf(x)> λ}).w,nZRn |f(x)| λ1+log+|f(x)| λn−1w(x)dx(1.6) whenever w∈A∗ ∞. This was proved by Jawerth and Torchinsky, [7], and independently by Long and Shen [10]. 4 T. LUQUE AND I. PARISSIS Fefferman-Stein inequality. By this we mean in general an inequality of the form ZRn (Mf)pw.w,nZRn |f|pMwdx, 1 < p < ∞, where Mdenotes some maximal operator. Inequalities of this type are important since, among other things, they can be used to derive the boundedness of vector-valued maximal operators. In fact this inequality was first proved for the Hardy-Littlewood maximal function by C. Fefferman and Stein, [4], for every non-negative, locally integrable weight w. The main application in [4] was exactly the vector-valued extension of the classical Hardy-Littlewood maximal theorem. For the strong maximal function the same inequality is true provided that w∈A∗ ∞; see [9] for a direct proof of this result and also [12], where the Fefferman-Stein inequality is obtained as a corollary of a more general two weight-norm inequality. Observe that, as in the case of (1.4), we need some extra assumption on the weight in order to prove the Fefferman-Stein inequality for the strong maximal function. This should be contrasted to the corresponding result for the Hardy-Littlewood weighted maximal function, as well as to the Fefferman-Stein inequality for the Hardy-Littlewood maximal function, where no assumption on the weight is needed. The form of the endpoint Fefferman-Stein inequality depends on the corresponding unweighted endpoint properties of the maximal operator under study. For the usual Hardy-Littlewood maximal function MQthe right statement is w({x∈Rn:MQf(x)> λ}).n 1 λZRn |f(x)|MQw(x)dx. The natural endpoint Fefferman-Stein inequality for the strong maximal function was proved by Mitsis, [11], in dimension n=2. In particular Mitsis showed that w({x∈R2:Mnf(x)> λ}).wZR2 |f(x)| λ1+log+|f(x)| λMnw(x)dx. The main result of the current paper is the extension of the endpoint Fefferman-Stein inequality for the strong maximal function to all dimensions: Theorem. Let w∈A∗ ∞. For all dimensions n⩾1we have w({x∈Rn:Mnf(x)> λ}).n,wZRn |f(x)| λ1+log+|f(x)| λn−1Mnw(x)dx. By interpolation, the Fefferman-Stein inequality of our main theorem above implies the strong Lp-version of the Fefferman-Stein inequality from [9], [12]. Furthermore, since every A∗ 1-weight is an A∗ ∞-weight, we recover the endpoint inequality (1.6) for A∗ 1-weights. It should be noted that the proof of Mitsis in [11] uses the combinatorics of two-dimensional rectangles, which allow one to get favorable estimates for the measures |{x∈Rk: N X k=1 1Rk(x) = ℓ}|; THE ENDPOINT FEFFERMAN-STEIN INEQUALITY FOR THE STRONG MAXIMAL FUNCTION 5 here {Rk}1⩽k⩽Nis a sequence of rectangles which satisfy a certain sparseness property and ℓis any integer in {1, 2, . . . , N}. These combinatorics do not seem to be readily available in higher dimensions and so we adopt a different approach, which relies on the boundedness of the weighted strong maximal function Mw nand the precise estimate for its norm, (1.5). In particular, our approach is inspired by the arguments in [10], a paper which seems to have been overlooked by most of the works on the weighted inequalities for the strong maximal function. Acknowledgments. This work was done while T.L was visiting T.Hytönen at University of Helsinki. The authors would like to thank him for his generosity and hospitality. We also want to thank C. Pérez and T. Mitsis for some valuable discussions on the subject of this paper. 2. Notation We write A.Bif A⩽CB for some numerical constant C > 0. In order to indicate the dependence of the constant on some parameter n(say), we write A.nB. Similarly, A≃B means that A.Band B.A. 3. Some geometry of n-dimensional rectangles In this section we recall some sparseness properties of n-dimensional rectangles, introduced in [3]. Here we adopt the slightly different approach from [10]. In fact, both Lemmas in this section are mentioned in [10]. However, we present the proofs for the sake of completeness. For t∈Rand E⊂Rnwe introduce the slice operator Pt(E)≔{x′∈Rn−1: (x′,t)∈E}. Thus Ptis the ‘slice’ of Eby a hyperplane perpendicular to the n-th coordinate axis at level t∈R. The (n−1)-dimensional projection is Pk(E)≔{x′∈Rn−1: (x′,t)∈Efor some t∈R}. We will also use the one-dimensional projection P⊥defined for E⊂Rnas P⊥(E)≔{t∈R: (x′,t)∈Efor some x′∈Rn−1}. If R∈Rnobserve that we have R=Pk(R)×P⊥(R) = Pt(R)×P⊥(R), for all t∈P⊥(R). For any interval I⊂R, let I∗be the interval with the same center and three times the length of I,|I∗|=3|I|. For R∈Rnwe then use the notation R∗≔Pk(R)×(P⊥(R))∗. Thus R∗is the rectangle with the same center as Rand whose sides parallel to the first n−1 coordinate axes have the same lengths as the corresponding sides of R; the side of Rwhich is parallel to the n-th coordinate axis has length equal to three times the length of the corresponding side of R. 6 T. LUQUE AND I. PARISSIS Let R={Rk}1⩽k⩽Nbe a finite sequence of rectangles from Rn. We will say that Rsatisfies the sparseness property (P2) if P⊥(R1)⩾P⊥(R2)⩾···⩾P⊥(RN), |Rk∩Sj<k R∗ j|⩽ǫ|Rk|,k=1, 2, . . ., N. (P2) Here 0 < ǫ < 1 will be assumed sufficiently small in various parts of the arguments below. For t∈Rwe now consider the collection T(t) = T={Pt(Rk)}1⩽k⩽N⊂Rn−1which is produced by slicing all the n-dimensional rectangles of Rby a hyperplane perpendicular to the n-th coordinate axis, at the level t. The collection Tdepends on tbut we will many times suppress this fact in what follows. The main point about the collections Rand Tis contained in the following standard fact. Lemma 3.1. Suppose that the sequence R={Rk}1⩽k⩽Nhas the sparseness property (P2). Then, for all t∈R, the (n−1)-dimensional collection of rectangles T(t) = {Pt(Rk)}1⩽k⩽Nhas the sparseness property (P1), uniformly in t: |Pt(Rk)∩[ j<k Pt(Rj)|⩽ǫ|Pt(Rk)|,k=1, 2, . . ., N.(P1) Proof. We fix some 1 ⩽k⩽Nand t∈P⊥(Rk). Denoting J≔{j < k :Pt(Rk)∩Pt(Rj),∅}we have by the second condition in (P2) that ǫ|Rk|⩾|Rk∩[ j<k R∗ j|=|[ j<k (Rk∩R∗ j)|⩾|[ j∈J (Rk∩R∗ j)|(3.2) Observe that for j∈Jwe have that ∅,P⊥(Rk)∩P⊥(Rj)∋tand by the first condition in (P2) we have |P⊥(Rj)|⩾|P⊥(Rk)|. A moment’s reflection shows that if I1,I2are intervals in R,|I2|⩾|I1|and I1∩I2,∅then I1⊆I∗ 2. We conclude that P⊥(Rk)⊆P⊥(R∗ j). Thus the n-dimensional rectangle Rk∩R∗ jis of the form P⊥(Rk)×Pk(Rk∩R∗ j). However, j∈Jimplies that Pt(Rk∩Rj) = Pt(Rk)∩Pt(Rj),∅, so we conclude that Pk(Rk∩R∗ j) = Pt(Rk∩Rj)and thus Rk∩R∗ j=P⊥(Rk)×Pt(Rk∩Rj).(3.3) Now estimate (3.2) and identity (3.3) give ǫ|P⊥(Rk)|×|Pt(Rk)|=ǫ|Rk|⩾[ j∈J P⊥(Rk)×Pt(Rk∩Rj) =|P⊥(Rk)|×[ j∈J Pt(Rk∩Rj) =|P⊥(Rk)|×Pt(Rk)∩[ j∈J Pt(Rj) =|P⊥(Rk)|×|Pt(Rk)∩[ j<k Pt(Rj)|. THE ENDPOINT FEFFERMAN-STEIN INEQUALITY FOR THE STRONG MAXIMAL FUNCTION 7 This proves the lemma for t∈P⊥(Rk)while for t<P⊥(Rk)the conclusion follows trivially.  The next lemma gives a precise quantitative bound on the overlap of the rectangles in Runder the sparseness property (P2). Lemma 3.4. Let w∈A∗ ∞and suppose that the finite sequence R={Rk}1⩽k⩽N⊂Rnsatisfies property (P2)with ǫsufficiently small, depending on the weight w. We set Ω≔∪N k=1Rk. For 1< p < ∞we have ZΩ N X k=1 1Rk pw(x)dx1 p .w,ncp,nw(Ω)1 p with cp,n=On(pn−1)as p→+∞. Proof. For a sequence {Rk}1⩽k⩽Nas before, consider the sequence T(t)of (n−1)-dimensional rectangles, by slicing the collection Rwith a hyperplane perpendicular to the n-th coordinate axis, at level t∈R. Let Ωt≔Pt(Ω)denote the corresponding slice of Ωat level tand set Tk≔Pt(Rk)in order to simplify the notation. By Lemma 3.1 the collection T(t) = {Tk}1⩽k⩽N has the property (P1). We set Ek≔Tk\∪j<kTj. For fixed t∈R, the function wt(x′) = w(x′,t),x′∈Rn−1, is an A∗ ∞-weight in Rn−1, uniformly in t∈R; see [5]. By the property (P1) and the fact that wt∈A∗ ∞uniformly in t, we will have that wt(Tk)⩾wt(Ek)⩾1 2wt(Tk)if ǫ > 0 was selected sufficiently small in property (P2), and thus also in (P1), according to Remark 1.3. Define the linear operator Lwtf(x′)≔ N X k=1 1 wt(Tk)ZTk f(y′)wt(y′)dy′1Ek(x′),x′∈Rn−1. For any locally integrable function fon Rn−1we have that Lwtf(x′)⩽Mwtf(x′),x′∈Rn−1. Also observe that for f,glocally integrable we have ZΩt Lwtf(x′)g(x′)wt(x′)dx′=ZΩt N X k=1 1 wt(Tk)ZEk g(y′)wt(y′)dy′1Rk(x′)f(x′)wt(x′)dx′ ≕ZΩt L∗ wtg(x′)f(x′)wt(x′)dx′. Furthermore L∗ wt(1Ωt) = PN k=1 wt(Ek) wt(Tk)1Tk⩾1 2PN k=11Tk. For any locally integrable function g on Rn−1we thus have ZΩt g(x′) N X k=1 1Tk(x′)wt(x′)dx′.ZΩt g(x′)L∗ wt(1Ωt)(x′)wt(x′)dx′ =ZΩt Lwtg(x′)wt(x′)dx′⩽kMwtgkLp′(wt,Rn−1)wt(Ωt)1 p .w,n(p′−1)−(n−1)kgkLp′(wt,Rn−1)wt(Ωt)1 p, 8 T. LUQUE AND I. PARISSIS by (1.5). Taking the supremum over g∈Lp′(Rn−1)with kgkLp′(wt,Rn−1)⩽1 gives the estimate ZΩt N X k=1 1Tk(x′) pwt(x′)dx .w,np(n−1)pwt(Ωt), as p→∞. It is essential to note here that this estimate is uniform in t∈R. Thus integrating over t∈P⊥(Ω), gives the claim.  4. Proof of the endpoint Fefferman-Stein inequality We begin with a simple lemma. Lemma 4.1. Let ǫ > 0,fbe a locally integrable function and set F≔{x∈Rn:Mnf(x)>1}. There exists a finite collection of rectangles R={Rs k}1⩽k⩽Nsuch that: (i) The collection Rhas the property (P2)with parameter ǫ. (ii) For each 1⩽k⩽Nwe have |Rs k|<ZRs k |f(y)|dy (iii) We have the estimate w(F).ǫ,w,nw(∪kRs k). Proof. For every x∈Flet Rx∈Rnbe a rectangle such that 1 |Rx|ZRx |f(y)|dy > 1. Without loss of generality we may assume that {Rx}x∈Fis a finite sequence {Rj}1⩽j⩽M, so that (ii) is satisfied and such that w(F)⩽w(∪1⩽j⩽MRj). From the collection {Rj}1⩽j⩽Mwe will now choose a subcollection {Rs k}1⩽k⩽Nso that (i) and (iii) are also satisfied. First we reorder the rectangles Rjso that P⊥(R1)⩾P⊥(R2)⩾···⩾P⊥(RM). We choose Rs 1≔R1and assume that the rectangles Rs 1,Rs 2,...,Rs τ, have been selected. Also let 1 ⩽jo< M so that Rs τ=Rjo. We then choose Rs τ+1to be the rectangle with the smallest index among the rectangles S∈{Rjo+1,...,RM} that satisfy |S∩[ j⩽τ (Rs j)∗|⩽ǫ|S|. Since the collection {Rj}1⩽j⩽Mis finite, the selection process will end after a finite number of N steps, and the collection {Rs k}1⩽k⩽Nwill automatically satisfy (i). Of course this subcollection still satisfies (ii). Now assume that some S∈{R1,...,RM}was not selected. We can then find some positive integer K∈{1, 2, . . ., N}such that |S∩[ j⩽K (Rs j)∗|> ǫ|S|. THE ENDPOINT FEFFERMAN-STEIN INEQUALITY FOR THE STRONG MAXIMAL FUNCTION 9 Thus we get for all x∈S Mn(1∪j⩽N(Rs j)∗)(x)⩾Mn(1∪j⩽K(Rs j)∗)(x)> ǫ, which means that [ 1⩽j⩽N Rjnot selected Rj⊆{x:Mn(1∪j⩽N(Rs j)∗)(x)> ǫ}. However, since w∈A∗ ∞we know that Mn:Lpo(w)→Lpo,∞(w)for some po>1. We conclude that w[ 1⩽j⩽N Rjnot selected Rj.ǫ,w,nw(∪j⩽N(Rs j)∗).w(∪j⩽NRs j). Thus w(F)⩽w(∪1⩽j⩽MRj).ǫ,w,nw(∪1⩽k⩽NRs k)as we wanted.  We are now ready to give the proof of our main result. Proof of the Main Theorem. We assume that n⩾2 since in one dimension M1is the usual Hardy-Littlewood maximal function and there is nothing (new) to prove. We henceforth write Mfor Mnsince the dimension nis fixed throughout the proof. Furthermore, it suffices to prove the theorem for λ=1. Let F≔{x∈Rn:M(f)(x)>1}and consider the collection R≔{Rs k}N k=1⊂Rngiven by Lemma 4.1. By (i) of that Lemma the collection Rhas the sparseness property (P2). We assume that ǫ > 0 was chosen small enough in Lemma 4.1, and thus in (P2), so that Lemma 3.4 is valid. Observe that (P2) also implies that Rs k∩[ j<k Rs j⩽ǫ|Rs k|. By choosing ǫ > 0 small enough we can also assume that w(Rs k∩Sj<k Rs j)⩽1 2w(Rs k), according to Remark 1.3. Setting Ek≔Rs k\Sj<k Rs jwe will thus have w(Rs k)⩾w(Ek)⩾1 2w(Rs k),k=1, 2, . . . , N,(4.2) and the choice of ǫ > 0 depends only on the weight w∈A∗ ∞. Denoting Ω≔SN k=1Rs k, we use (ii) and (iii) of Lemma 4.1 to estimate w(F).ǫ,w,nw(Ω)⩽ N X k=1 w(Rs k)⩽ N X k=1 w(Rs k) |Rs k|ZRs k |f(y)|dy =ZΩ f(x) N X k=1 w(Rs k) |Rs k|1Rs k(x)dx.