Sharp weighted inequalities for the vector-valued maximal function

Trans. Amer. Math. Soc. 352 (2000), 3265–3288. Sharp weighted inequalities for the vector–valued maximal function Carlos P´erez Departmento de Matem´aticas Universidad Aut´onoma de Madrid 28049 Madrid, Spain e–mail: carlos.p[email protected] work partially supported by DGICYT grant PB940192, Spain 1 1 Motivation and description of the main results The purpose of this paper is to obtain some sharp weighted inequalities for the vector–valued maximal function Mqwhich are not within the scope of the standard Aptheory for vector–valued singular integrals as can be found in [RRT]. We start with a review of some of the classical estimates and then we shall state the main results. 1.1 Background Let Mbe the Hardy–Littlewood maximal function and let Mqbe the vector–valued maximal operator defined by Mqf(x) = ∞ X i=1 (Mfi(x))q!1/q . This nonlinear operator was introduced by C. Fefferman and E. M. Stein in [FS] as a generalization of both the (scalar) maximal function Mand the classical integral of Marcinkiewicz and since then it has played an important role in the development of modern Harmonic Analysis. We recall the two basic estimates obtained in [FS] for 1 < q < ∞: •Let 1 < p < ∞, then there exists a constant Csuch that ZRnMqf(x)pdx ≤CZRn|f(x)|p qdx. (1) •The following weak type (1,1) estimate holds: there exists a constant csuch that sup λ>0 λ{x∈Rn:Mqf(x)> λ}≤CZRn|f(x)|qdx. (2) We are using here the notation |f(x)|q= (P∞ i=1 |fi(x)|q)1/q =kf(x)k`q. Another fundamental generalization of the maximal theorem is due to B. Muckenhoupt [M] who gave a characterization of following “weighted norm inequality” ZRnMf(x)pw(x)dx ≤cZRn|f(x)|pw(x)dx, (3) in terms of the Apcondition of Muckenhoupt: there exists a positive constant csuch that for all cubes Q Ap 1 |Q|ZQ w(y)dy 1 |Q|ZQ w(y)1−p0dyp−1 ≤c. (4) It is also well known that the Apcondition (4) also characterizes all the weights wfor which the weighted vector–valued inequality holds ZRnMqf(x)pw(x)dx ≤CZRn|f(x)|p qw(x)dx. (5) This result is due to K. F. Andersen and R. T. John [AJ], and to V. Kokilashvili [K]. There are by now three ways of proving (5): 2 •Refining the argument of Fefferman and E. M. Stein in [FS] as done in [AJ] and [K]. •Looking at Mqas a vector–valued singular integral with operator–valued kernel satisfying a pointwise gradient condition as can be found in [RRT]. •By applying the extrapolation theory of J. Garcia-Cuerva and J. L. Rubio de Francia as mentioned in [GCRdF] p. 521 which yields a simple proof. In this paper we investigate the two weight problem for the vector-valued maximal function Mq ZRnMqf(x)pw(x)dx ≤CZRn|f(x)|p qv(x)dx, (6) for which none of the above approaches works. Recall that there is a characterization due to E. Sawyer [S] of the two weight problem in the scalar situation, the following weighted inequality: ZRnMf(x)pw(x)dx ≤CZRn|f(x)|pv(x)dx, (7) holds if and only if there exists a constant csuch that for all cubes Q SpZQ M(v1−p0χQ)(y)pw(y)dy ≤cZQ v(y)1−p0dy. (8) The range p≤qfor (6) is easy to handle since it coincides with the scalar situation. Indeed, if p≤qwe claim that the Spcondition is necessary and sufficient for (6). It is clear that condition (8) is necessary. If we assume (8) we see that (6) is immediate for both q=pand q=∞. Then the case 1 <p<q<∞follows by interpolation for “linearizable operators” in the vector–valued context (cf. the argument given in [GCRdF] p. 482). Although we shall give a full characterization of (6) in Theorem 2.3 we are more interested in estimates of the form ZRnMqf(x)pw(x)dx ≤CZRn|f(x)|p qNw(x)dx, (9) where Nis an appropiate (scalar) maximal type operator. Needless to say that the prototypical estimate that we have in mind is the Fefferman–Stein weighted inequality ZRnMf(x)pw(x)dx ≤CZRn|f(x)|pMw(x)dx, (10) which yields, as it is well known, the unweighted vector–valued estimate (1) when p>q. One of the main observations that follows from our results is that Mqdoes not verify a similar inequality to (10) on the range p > q (cf. the first remark after Theorem 1.1). Inequalities of the type (10) reflect how singular is the operator under study. This can be seen for instance with the following sharp inequalities for singular integrals obtained in [P2] for p > 1 generalizing some previous estimates obtained by M. Wilson in the range 1 < p ≤2 [Wil2]: Let Tbe any Calder´on–Zygmund operator, and let 1 <p<∞. Then there exists a constant Csuch that 3 ZRn|Tf(x)|pw(x)dx ≤CZRn|f(x)|pM[p]+1w(x)dx, (11) with Cindependent of wand f. Furthermore, the estimate is sharp since it does not hold for M[p]. Here Mk=M◦(k) . . . ◦M k = 1,2,· · ·, denotes the Hardy–Littlewood maximal operator Miterated ktimes. Another example which stresses our point of view is related to the classical Area function. This non linear operator is defined by the integral Sϕ(f)(x) = ZBt(x) |f∗ϕt(y)|2dt dy tn+1 !1/2 , where ϕ∈C∞ 0with Rϕ= 0 and ϕt(x) = t−nϕ(x t), t > 0. Then the Area function satisfies the following inequality: Let 1 < p ≤2, then there exists a constant Csuch that ZRnSϕ(f)(x)pw(x)dx ≤CZRn|f(x)|pMw(x)dx. (12) Furthermore, this inequality is false for p > 2. The case p= 2 was first obtained by A. Chang, M. Wilson, and T. Wolff in [CWW] and for 1<p<2 by S. Chanillo and R. Wheeden in [CW] as well as the counterexample for p > 2. See also the work by M. Wilson [Wil1]–[Wil4]. 1.2 Main results Motivated by the Theorems mentioned above we state now the main result of the paper. THEOREM 1.1 Let 1< q < p < ∞. a)There exists a constant Csuch that ZRnMqf(x)pw(x)dx ≤CZRn|f(x)|p qM[p q]+1w(x)dx, (13) for all locally integrable functions f,w≥0. b) Part a) is sharp since there exists no constant Csuch that ZRnMqf(x)pw(x)dx ≤CZRn|f(x)|p qM[p q]w(x)dx, (14) for all locally integrable functions f,w≥0. Likewise, the corresponding weak type (p, p)estimate is false. We now make the following remarks. (a) It follows from part b) of the Theorem that the vector-valued analogue of the Fefferman-Stein inequality (10) 4 ZRnMqf(x)pw(x)dx ≤CZRn|f(x)|p qMw(x)dx. (15) is false in general in the range p > q. (b) If we look at the proof of (13) we see that we can refine such an inequality by replacing M[p q]+1 by ML(log L) p q−1+, > 0, or by MAwhere Asatisfies Z∞ ct A(t)(p q)0−1dt t<∞. See Section 3 for the appropiate definition of the maximal type function MA. (c) We emphasize on the fact that there is no assumption on wother than local integrabilty. In fact, if we assume that w∈A∞then (15) holds being false in general. Indeed, by the Lebesgue differentation Theorem we have ZRnMqf(x)pw(x)dx ≤ZRnMqf(x)pMw(x)dx ≤CZRn|f(x)|p qMw(x)dx, (16) where in the last inequality we have used the Apresult for Mq(5) since Mw ∈A1by standard results (see the last part of Section 4). Also, we may replace M[p q]+1wby the A1weight M(wr)(x)1/r,r > 1, by applying again (5). However, the later class of weights (essentially the class A1) are pointwise larger than the non A∞ weights Mkwsince it may be shown using standard theory that For each integer k= 1,2,· · ·, each r > 1 and each locally integrable function f, we have the following pointwise inequality for all x∈Rn: w(x)≤Mkw(x)≤[M(wr)1/r]k−1 A1M(wr)(x)1/r. Here [w]A1denotes the “norm” of w∈A1, namely the smallest constant Csuch that Mw ≤C w. (d) Theorem 1.1 indicates that Mqbehaves more as a singular integral operator rather than as a maximal operator. However, we want to emphasize the fact that inequality (13) does not fit within the scope of the theory of vector–valued singular integrals as developed by J. L. Rubio de Francia, F. J. Ruiz and J. L. Torrea in [RRT] where the pioneering work [BCP] was updated. In [RRT], the operator Mq, as well as many other non–linear operators such as the Area function Sϕ, are seen as singular integrals taking values in an appropiate Banach space. Using this point of view, it is possible to translate to this more general context the one weight scalar Aptheory at least for any vector–valued singular integral with sufficiently smooth kernel. However, this is not the case of (13) (nor of (12)) since the result for the (scalar) Hilbert transform (11) is worse than (13) indicating that the operator Mqis less singular than H. The proof of the positive part of Theorem 1.1 does not follow the scheme used in [P2] to treat singular integrals since we cannot dualize (13). We shall derive (13) as a consequence of a 5 characterization for the two weight problem given in Theorem 2.3. The condition we obtain is a blend of Sawyer’s condition Sptogether with Rubio de Francia’s characterization of vector– valued inequalities for sublinear operators as can be found in [GCRdF] Chapter VI. It should be mentioned that Y. Rakotondratsimba has obtained in [R] a different characterization of the two weight problem which is much closer in spirit to Sawyer’s condition Sp. 1.3 Sharp sufficient conditions close to Ap In this section we take up the two weight problem for Mqthat we write in the following more convenient form ZRn(w(x)Mqf(x))pdx ≤CZRn(v(x)|f(x)|q)pdx. (17) The task is to provide sharp sufficient conditions on the weights “close” in structure to the Ap condition. Let us briefly review some results related to the scalar situation ZRn(w(x)Mf(x))pdx ≤cZRn(v(x)|f(x)|)pdx. (18) It is well known that the necessary Apcondition for this problem 1 |Q|ZQ w(x)pdx1/p 1 |Q|ZQ v(x)−p0dx1/p0 ≤c(19) is not sufficient, and that the correct necessary and sufficient condition is, as we mentioned above, Sawyer’s condition which with our normalization on the weights has the following form ZQ (w(y)M(v−p0χQ)(y))pdy ≤CZQ v(y)−p0dy. The drawback of this condition is that it involves the operator Mitself, and it would be interesting to obtain sufficient conditions close in form to the Apcondition (19). Perhaps, the first result in that direction was obtained by C. Neugebauer in [N]. He noticed that if (w, v) is a couple of weights such that for some r > 1 1 |Q|ZQ w(y)pr dy1/pr 1 |Q|ZQ v(y)−p0rdy1/p0r ≤c(20) for all cubes Q, then ZRn(w(y)Mf(y))pdy ≤cZRn(v(y)|f(y)|)pdy. (21) In fact Neugebauer proves that (20) is equivalent to showing that there is an Apweight inserted (pointwise) between wpand vpand the result follows trivially. This problem has been considered in [P1] where it is shown that such a strong condition is not needed. In particular it is not necessary to “bump” the left weight wand that much less than a power “bump” is required on the right weight vto get the result. We extract the following result from [P1]. Recall that for a given Young function Aand a cube Qon Rnwe defined the A-average of a function fover Qby kfkA,Q = inf{λ > 0 : 1 |Q|ZQ A|f(y)| λdy ≤1}. 6 THEOREM 1.2 [P1] Let 1<p<∞, and let Bbe a doubling Young function such that Z∞ c tp0 B(t)!p−1dt t<∞,(22) for some positive constant c. Let (w, v)be a couple of weights such that there is a positive constant Kfor which 1 |Q|ZQ w(y)pdy1/p  v−1 B,Q ≤K, (23) for all cubes Q. Then ZRn(w(y)Mf(y))pdy ≤cZRn(v(y)f(y))pdy (24) for all nonnegative functions f. As we may expect we need to consider stronger conditions on the weights to get corresponding results for Mqin the range p > q. In particular we need to “bump” the left weight was well since otherwise the result is false as the counterexample (w, Mw) in (15) shows. Indeed, observe that this pair of weights satsfies (23) for any cube Qand any Young function B: 1 |Q|ZQ w1/p   (Mw)−1/p  B,Q ≤1 |Q|ZQ w1/p     1 |Q|RQw−1/p    B,Q =1 |Q|ZQ w1/p 1 |Q|ZQ w−1/p k1kB,Q = 1 since 1 |Q|RQw≤Mw(x) for x∈Q. THEOREM 1.3 Let 1<q<p<∞, and let r=p q. Let A, B be doubling Young functions such that both Z∞ ctr A(t)r0−1dt tand Z∞ c tq0 B(t)!q−1dt t,(25) are finite for some positive constant c, that is ¯ A∈Br0and ¯ B∈Bq. Let (w, v)be a couple of weights such that there is a positive constant Kfor which kwqk1/q A,Q  v−1 B,Q ≤K, (26) for all cubes Q. Then the two weighted vector–valued inequality   (P∞ i=0(w Mfi)q)1/q  Lp(Rn)≤C  (P∞ i=0 |v fi|q)1/q  Lp(Rn)(27) holds for all fi. Some interesting examples are given by A(t)≈tr(log t)r−1+δand B(t)≈tq0(log t)q0−1+δwith δ > 0. 7 1.4 Endpoint estimates Although the operator Mqis, to some extent, more closely related to a singular integral this is not the case when we look at endpoint estimates such as the following. THEOREM 1.4 There exists a constant Csuch that for each weight wand for all λ > 0 w({x∈Rn:Mqf(x)> λ})≤C λZRn|f(x)|qMw(x)dx. (28) This result reflects once again that sharp results for Mqare independent from the theory of vector– valued singular integrals since we do not know whether the (scalar) Hilbert transform H satisfies w({x∈Rn:|Hf(x)|> λ})≤C λZRn|f(x)|Mw(x)dx. (29) See [P2] for sharp results. There exists an interesting relationship between (28) and a possible vector–valued version of the classical Besicovitch lemma. We shall formulate this as a conjecture. Mc wdenotes the weighted centered maximal function. CONJECTURE 1.5 w({x∈Rn: ∞ X i=1 (Mc wfi(x))q!1/q > λ})≤C λZRn|f(x)|qw(x)dx. (30) One can show (cf. Section 6) that if the conjecture were true then the inequality (28) follows immediately. Acknowledgements. The author is very grateful to A. Vargas for several conversations concerning the problems considered in this paper. 2 A characterization of the two weight problem The purpose of this section is to give a characterization of the two weight problem for the vector– valued maximal function Mq. We recall that the case 1 < p ≤qis characterized by means of Sawyer’s condition Sp. The main result is Theorem 2.3. For the proof of this Theorem it will be more efficient to work within a more general context. Let Bbe a basis in Rn, and by this we mean a collection of open sets in Rn. We say that wis a weight associated to the basis Bif wis a non-negative measurable function in Rnsuch that w(B) = RBw(y)dy < ∞for each Bin B.MB,w is the corresponding maximal operator defined by MB,wf(x) = sup x∈B 1 w(B)ZB |f(y)|w(y)dy if x∈ ∪B∈B and MB,wf(x) = 0 otherwise. If w≡1, we just write MBf(x). 8 PROPOSITION 2.1 Let 1< q < p < ∞, and let r=p q. Suppose that MB,σ :Lp `q(σ)→Lp `q(σ)where σ=v1−p0. Then the two weight vector valued inequality   (P∞ i=0(MBfi)q)1/q  Lp(w)≤C  (P∞ i=0 |fi|q)1/q  Lp(v)(31) holds if and only if there exists a constant csuch that for each g∈Lr0(Rn)we can find G∈Lr0(Rn) with kGkLr0(Rn)≤ kgkLr0(Rn)such that ZΩ MB(σχΩ)(x)qw(x)1/r g(x)dx ≤cZΩ σ(x)1/r G(x)dx, (32) for every set Ωwhich is a union of sets in B. Proof: We first show that condition (32) is necessary. First observe that inequality (31) is equivalent to    P∞ i=0(w1/p MB(fi v1/p ))q1/q   Lp(Rn) ≤c  (P∞ i=0 |fi|q)1/q  Lp(Rn). Now, by Rubio de Francia’s theorem (cf. [GCRdF] p. 555) this estimate is equivalent to showing that for each g∈Lr0(Rn) there exists G∈Lr0(Rn) with kGkLr0(Rn)≤ kgkLr0(Rn)and ZRn(w(y)1/pMB(f v1/p )(y))qg(y)dy ≤CZRn|f(y)|qG(y)dy, for all f, or what is the same ZRnMB(f)(y)qw(y)1/rg(y)dy ≤CZRn|f(y)|qv(y)1/rG(y)dy, for all f. Testing this inequality with f=σχΩ=v1−p0χΩgives the necessary condition (32). To prove the sufficiency of (32) we use that Lrand Lr0are dual spaces. We adapt the basic ideas from [GCRdF]. If we define Ias I=   P∞ i=0(w1/p MBfi)q1/q    q Lp(Rn) , then I= ∞ X i=0 ZRnMB(fi)(y)qw(y)1/r g(y)dy for some g∈Lr0(Rn) with unit norm. Fix i, and for each integer kconsider the set Ei k={y∈ Rn: 2k< MBfi(y)≤2k+1}. From the definition of MB,Ei k⊂ ∪jBi k,j, where Bi k,j ∈ B satisfies 2k<1 Bi k,jZBi k,j fi(y)dy. Define now Ei k,1=Bi k,1∩Ei k,and for j > 1Ei k,j =Bi k,j\ ∪s<j Bi k,s∩Ei k. For any fixed k, each of the sets Ei kis the disjoint union of the sets Ei k,j. We now can write 9 We also postpone the proof of this lemma until the end of the proof of the theorem. Now, using (49) and (51) we can estimate the left side of (43) as follows ZRnMBf(y)pw(y)dy =X kZΩk−Ωk+1 MBf(y)pw(y)dy (52) ≤apX k akpw(Ωk)≤CX k,j akpw(3Qk,j)≤ ≤CX k,j kfkp B,Qk,j w(3Qk,j) = CX k,j kfkp B,Qk,j w(3Qk,j) |3Qk,j||Qk,j| ≤CX k,j      fw(3Qk,j ) |3Qk,j|1/p     p B,Qk,j |Ek,j| ≤CX k,j ZEk,j MB(f(Mw)1/p)(y)pdy ≤CZRnMB(f(Mw)1/p)(y)pdy ≤CZRnf(y)pMw(y)dy, since we are assuming ii). This proves iii). Let us assume that iii) holds. Observe that (44) is equivalent with ZRnM(fg)(y)pw(y) [M¯ B(g)(y)]pdy ≤cZRnf(y)pMw(y)dy, for all nonnegative functions f,g, and w. Then iv) follows immediately from (43) after an application of the inequality M(fg)(y)≤MBf(y)M¯ Bg(y)y∈Rn which is a consequence of the generalized H¨older’s inequality (41). To prove that iv) implies i) we let w= 1 in (44) obtaining ZRnMf(y)p1 [M¯ B(u1/p)(y)]pdy ≤cZRnf(y)p1 u(y)dy, for all nonnegative functions f, and u. Testing this inequality with f=u=χQ(0,1) , where Q(x, r) denotes the cube centered at x∈Rnand with sidelength equal to r, we have ZRnMf(y)p1 [M¯ B(f)(y)]pdy ≤C. (53) On the other hand we have for large xthat M¯ B(f)(x)≈1 ¯ B−1(1 |x|n). Therefore we get 16 ZRnMf(y)p1 [M¯ B(f)(y)]pdy ≥CZ|y|>c 1 |y|np 1 ¯ B−1(1 |y|n)pdy =CZ∞ c 1 rnp 1 ¯ B−1(1 rn)prndr r≈Z∞ c B(t) tp dt t. This estimate combined with (53) shows that iv) ⇒i). To conclude the proof of the Theorem, apart from the proofs of Lemmas 3.3 and 3.4, we need to show that i) ⇔v). That i) is necessary is trivial since v) implies the scalar case, namely ii). To show that i) is sufficient observe that the case p<sfollows by interpolation from the cases s=p, s=∞. Now the case p > s follows from the weighted inequality iii) by standard arguments. 2 Proof of Lemma 3.3: The proof is a simple adaptation of arguments in [GCRdF] Ch. 2. Since fis bounded with compact support, say suppf⊂K, kfkB,Q ≤ kfkL∞kχKkB,Q =kfkL∞ 1 B−1|Q| |Q∩K|, and it follows that kfkB,Q →0 as Q↑Rn. Hence, if there are any dyadic cubes Qwith kfkB,Q > t, they are contained in cubes of this type which are maximal with respect to inclusion. We let Ct={Pj}be the family of the dyadic maximal nonoverlapping cubes satisfying t < kfkB,Pj. Let P0 jbe the only dyadic cube containing Pjwith sidelength twice that of Pj. Then t < kfkB,Pj≤2nkfkB,P 0 j . The last inequality can easily be deduced from the definition of the Luxemburg norm using the fact that t→B(t) tis non decreasing. Hence by the maximality of the cubes {Pj}we get t < kfkB,Pj≤2nt. (54) Observe that from this discussion it is clear that {y∈Rn:Md Bf(y)> t}=∪jPj.(55) Let x∈Ωt. By definition, there is a cube Rcontaining xsuch that t < kfkB,R .(56) 17 Let kbe the unique integer such that 2−(k+1)n<|R| ≤ 2−kn. There is some dyadic cube with side length 2−k, and at most 2nof them, {Ji:i= 1, . . . , n}, meet the interior of R. It is easy to see that for one of these cubes, say J1,t 2n<  χJ1f  B,R .(57) This can be seen as follows. If for each i= 1, . . . , 2nwe had   χJif  B,R ≤t 2n, we would get since R⊂ ∪2n i=1Jithat kfkB,R =    χ∪2n i=1Jif   B,R ≤ 2n X i=1   χJif  B,R ≤2nt 2n=t, contradicting (56). Since |R| ≤ |J1|<2n|R|one can also show t 4n<kfkB,J1.(58) By letting Ct/(4)n={Qj}, we have by (54) that t 4n<kfkB,Qj≤t 2n,(59) for each j, yielding (46). Equation 48) also follows since {y∈Rn:Md Bf(y)>t 4n}=∪jQj. Also, we see from (58) that J1⊂Qk, for some k, and then R⊂3J1⊂3Qk. This gives Ωt⊂ ∪j3Qj, which is (45). Now, by the left side of the inequality (59), and the definition of kfkB,Q we get |Ωt| ≤ CX j |Qj| ≤CX jZQj B4nf(y) tdy ≤CZRnBf(y) tdy. (60) To obtain (47) we just use the standard idea of writing fas f=f1+f2, where f1(x) = f(x) if f(x)>t 2, and f1(x) = 0 otherwise. Then MBf(x)≤MBf1(x) + MBf2(x)≤MBf1(x) + t 2. Finally, since (60) holds for each f≥0, t > 0 we have |Ωt| ≤ {y∈Rn:MBf1(y)>t 2}≤CZRnBf1(y) tdy =CZ{y∈Rn:f(y)>t/2} Bf(y) tdy, concluding the proof of Lemma 3.3. 2 18 We now conclude the proof of the Theorem by proving Lemma 3.4. Proof of Lemma 3.4: The family Ek,j is clearly disjoint. We note that (49) and the definition of the Luxemburg norm implies that 1<1 |Qk,j|ZQk,j B4n akf(y)dy, and 1 |Qk,j|ZQk,j B2n akf(y)dy ≤1. Hence by standard properties of the dyadic cubes we can estimate what portion of Qk,j is covered by Dk+1 as in [GCRdF] p. 398 |Qk,j ∩Dk+1| |Qk,j|=X i |Qk,j ∩Qk+1,i| |Qk,j|=X i:Qk+1,i⊂Qk,j |Qk+1,i| |Qk,j| <X i:Qk+1,i⊂Qk,j 1 |Qk,j|ZQk+1,i B4n ak+1 f(y)dy ≤2n a 1 |Qk,j|ZQk,j ∩∪iQk+1,i B2n akf(y)dy ≤2n a. Here we have used that B(2n at)≤2n aB(t), t > 0, since 2n a<1, and because t→B(t) tis increasing. This gives (50). Finally |Ek,j| |Qk,j|>1−2n a>0, completing the proof of the Lemma and hence that of Theorem 3.2. 2 4 Proof of the main Theorem In this section we give the proof of Theorem 1.1. We start with the proof of part a), the positive part. We apply Theorem 2.3 by verifying condition (37). The weight wis fixed and vwill be chosen ina a moment. Recall that r=p qand that σ=v1−p0. We need to show that there exists a constant csuch that for arbitrary g∈Lr0(Rn) there is G∈Lr0(Rn) with kGkLr0(Rn)≤ kgkLr0(Rn)and such that for all cubes Q ZQ M(σχQ)(x)qw(x)1/rg(x)dx ≤cZQ σ(x)1 rG(x)dx. (61) We use the Fefferman-Stein inequality (10) together with the generalized H¨older’s inequality (41) to estimate the left hand side of (61) by a multiple of ZQ σ(x)qM(w1/rg)(x)dx ≤ZQ σ(x)qMB(w1/r)(x)M¯ B(g)(x)dx =ZQ σ(x)q[(MB(w1/r)(x))r]1/rM¯ B(g)(x)dx. 19 If we let v= (MB(w1/r))rwe have that ZQ M(σχQ)(x)qw(x)1/rg(x)dx ≤CZQ σ(x)1/r M¯ B(g)(x)dx. To conlude the proof all we have to do know is to choose Bsuch that ¯ B∈Br0, namely that Z∞ ctr B(t)r0−1dt t<∞ since by the characterization in Theorem 3.2 M¯ B:Lr0(Rn)→Lr0(Rn). If we let  M¯ B be the norm of this operation we can take G=M¯ B(g)  M¯ B  such that kGkLr0(Rn)≤ kgkLr0(Rn)and we have ZQ M(σχQ)(x)qw(x)1/rg(x)dx ≤C M¯ B(g) ZQ σ(x)1/r G(x)dx. Finally we are left with showing that we can pick Bsuch that v= (MB(w1/r))r≤M[r]+1w. Indeed, let B(t)≈tr(log(1 + t))[r], then B(t1/r)≈t(log(1 + t)[r]and (MB(w1/r))r=ML(log L)[r](w). Now, we make the following observation. Let k= 1,2,3,· · ·, then there exists a constant C=Cnsuch that for all bounded functions fwith support contained in Q kfkL(log L)k,Q ≤C |Q|ZQ Mkf(y)dy. Indeed, by homogeneity we can assume that the right hand side is one. Then by the definition of the Luxemburg norm it is enough to prove 1 |Q|ZQ f(y)(1 + log+(f(y)))kdy ≤C, which is a consequence of iterating the following well known inequality of E.M. Stein: ZQ f(y)(1 + log+(f(y)))kdy ≤CZQ Mf(y)(1 + log+(Mf(y)))k−1dy, (62) with k= 1,2,3,· · ·. Therefore we finally have that v=ML(log L)[r](w)≤M[r]+1wconcluding the proof of the Theorem. This concludes the proof of the first part, for the counterexample we take n= 1 and we let N be a large positive integer and r=p q>1. Set w=χ(0,1) and define fi(x) = (log x)−1/qχ(ei,ei+1)(x) for each i= 1,· · · , N −1, and fi= 0 for i≥N. Then,   (P∞ i=1 |fi|q)1/q   p Lp(M[r]w)=ZR N−1 X i=1 (log x)−1χ(ei,ei+1)(x)!r M[r]w(x)dx 20 ≈ZeN e (log x)−r(log x)[r]−1dx x. When ris an integer this is comparable to log N, and when ris not an integer the integral is a constant independent of Nsince [r]−r+ 1 >0. In any case, it is less than a constant times log N. On the other hand   (P∞ i=1(Mfi)q)1/q   p Lp(w)=Z1 0 N−1 X i=1 (Mfi(x))q!r dx ≥Z1 0 N−1 X i=1 1 i!r dx ≈(log N)r, since for 0 < x < 1 and i= 1,· · · , N −1, Mfi(x)≥1 ei+1 Zei+1 0 (log y)−1/qχ(ei,ei+1)(y)dy ≈1 i1/q . To conclude, observe that (log N)r≤Clog Ndoes not make sense for large Nwith Cindependent of N.2 We conclude the section by giving the following simple argument showing that Mw ∈A1 assuming that w∈A∞which was used to prove inequality (16). Indeed since wstaisfies for some r > 1 the reverse H¨older inequality 1 |Q|ZQ wr1/r ≤C |Q|ZQ w with Cindependent of the cube Q. Now for fixed Qand x∈Qwe have 1 |Q|ZQ Mw ≤1 |Q|ZQ M(wχ2Q) + 1 |Q|ZQ M(wχRn\2Q) ≤1 |Q|ZQ M(wχ2Q)r1/r +Cinf QM(w)≤C1 |2Q|Z2Q wr1/r +M(w)(x) ≤C |2Q|Z2Q w+M(w)(x)≤C M(w)(x). Here we have used that M(χRn\2Qw)(y)≈M(χRn\2Qw)(z) for each y, z ∈Q, [GCRdF] p. 159 and the Lrboundedness of M. This means that Mw ∈A1. 5 Proof of the sharp sufficient conditions We want to point out that we do not know how to prove this theorem directly from the characterization given in Theorem 2.3. We are going to modify and combine the proof of this Theorem with the results in Theorem 3.2 which in fact contains the key estimate. Proof of Theorem 1.3: 21   (P∞ i=0(wMfi)q)1/q   q Lp(Rn)= ∞ X i=0 ZRnMfi(y)qw(y)qg(y)dy (63) for some g∈Lr0(Rn) with unit norm. Let ibe fixed. For each integer k, and for any arbitrary constant a > 2nwe let Ωi k, and Di kbe the sets Ωi k={x∈Rn:ak< Mfi(x)}, Di k={x∈Rn:Mdfi(x)>ak 4n}. By the classical Calder´on–Zygmund decomposition (cf. [GCRdF] p. 137) there exists a family of maximal nonoverlapping dyadic cubes {Qi k,j}for which Ωi k⊂ ∪j3Qi k,j,Di k=∪jQi k,j, and ak 4n<1 Qi k,jZQi k,j fi(y)dy ≤ak 2n.(64) We can now estimate the integral in (63) as follows ZRnMfi(y)qw(y)qg(y)dy =X kZΩi k−Ωi k+1 Mfi(y)qw(y)qg(y)dy (65) ≤aqX k akq(wqg)(Ωi k)≤CX k,j akq(wqg)(3Qi k,j) ≤CX k,j   1 Qi k,jZQi k,j fi(y)dy  q (wqg)(3Qi k,j) =CX k,j   1 Qi k,jZQi k,j fi(y)v(y)v(y)−1dy  q (wqg)(3Qi k,j) ≤CX k,j   1 3Qi k,jZ3Qi k,j fi(y)v(y)v(y)−1dy  q1 3Qi k,jZ3Qi k,j w(y)qg(y)dy Qi k,j. For each integer k, j we set Ei k,j =Qi k,j −Qi k,j ∩Di k+1. Then {Ei k,j}is a disjoint family of sets, and by Lemma 3.3 with B(t) = t, there is a positive constant βsuch that for each k, j Qi k,j< β Ei k,j. This together with the generalized H¨older’s inequality (41) allows to dominate the last sum by a multiple of X k,j kvfikq ¯ B,3Qi k,j  v−1  q B,3Qi k,j kwqkA,3Qi k,j kgk¯ A,3Qi k,j Ei k,j ≤KqX k,j ZEi k,j M¯ B(vfi)(y)qM¯ A(g)(y)dy ≤CZRnM¯ B(vfi)(y)qM¯ A(g)(y)dy. since the sets {Ei k,j}are pairwise disjoint when iis fixed. Hence, by H¨older’s inequality with exponents rand r0we can estimate (63) by 22   (P∞ i=0(wMfi)q)1/q   q Lp(Rn)≤C  P∞ i=0(M¯ B(fiv))q1/q   q Lp(Rn) M¯ A(g) Lr0(Rn) ≤C  (P∞ i=0(vfi)q)1/q   q Lp(Rn)kgkLr0(Rn)=  (P∞ i=0(vfi)q)1/q   q Lp(Rn) since M¯ B:Lp `q(Rn)→Lp `q(Rn) and M¯ A:Lr0(Rn)→Lr0(Rn) by Theorem 3.2. 2 6 Endpoint estimates and the Besicovitch lemma Proof of Theorem 1.4: It is enough to consider f≥0 in the sense that fi≥0 for all i. Let Ω = {x∈Rn:Md(|f|q)(x)> λ}=∪Q, where the dyadic cubes Qare maximal nonoverlapping satisfying λ < 1 |Q|ZQ |f(x)|qdx ≤2nλ, (66) and |f(x)|q≤λ, a.e. x∈Rn\Ω. Write f=f·χRn−Ω+f·χΩ=g+b. Since Mqf≤Mqg+Mqb, it is sufficient to estimate the distribution set of Mqgand Mqbseparetly. Observing that |g(x)|q≤λwe get by (10) w({x∈Rn:Mqg(x)> λ/2})≤C λqZRnMqg(x)qw(x)dx ≤C λqX iZRn(Mgi)(x)qw(x)dx ≤C λqX iZRngi(x)qMw(x)dx =C λqZRn|g(x)|q qMw(x)dx ≤C λZRn|f(x)|qMw(x)dx. For bwe split the distibution set of Mqbset as follows. Let ˜ Ω = ∪˜ Q,˜ Q= 3Q. Then w({x∈Rn:Mqb(x)> λ/2})≤w({x∈Rn\˜ Ω : Mqb(x)> λ/2}) + w(˜ Ω). The second term is estimated by the left hand side of (66): w(˜ Ω) ≤C λX Q w(˜ Q)1 |Q|ZQ |f(x)|qdx ≤C λX QZQ |f(x)|qMw(x)dx ≤C λZRn|f(x)|qMw(x)dx. For the first term we use the argument in [FS] p. 110 which shows that Mbi(x)≤M(¯ bi)(x), x∈Rn\˜ Ω where ¯ biis the function 23 ¯ bi(x) = n0x∈Rn\Ω1 |Q|RQfi(y)dy x ∈Q w({x∈Rn\˜ Ω : Mqb(x)> λ/2} ≤ 1 λqX iZRn\˜ Ω (Mbi(x))qw(x)dx ≤1 λqX iZRn\˜ Ω (M(¯ bi)(x))qw(x)dx ≤C λqX iZRn ¯ bi(x)qM(wχRn\˜ Ω)(x)dx =C λqX iX Q1 |Q|ZQ fi(y)dyqZQ M(wχRn\˜ Q)(x)dx ≤C λqX Q"X i1 |Q|ZQ fi(y)dyq#1 qqZQ M(wχRn\˜ Q)(x)dx ≤C λqX Q1 |Q|ZQ |f(y)|qdyq inf QM(w)|Q| ≤ C λX Q 1 |Q|ZQ |f(y)|qdy inf QM(w)|Q| ≤C λX QZQ |f(x)|qMw(x)dx ≤C λZRn|f(x)|qMw(x)dx. 2 As we mentioned in the introduction there exists a close connection between the weighted scalar inequality w({x∈Rn:Mf(x)> λ})≤C λZRn|f(x)|Mw(x)dx (67) and the the Besicovitch covering lemma [dG]. Indeed, the first observation is that (67) is equivalent to w({x∈Rn:M(fw Mw)(x)> λ})≤C λZRn|f(x)|w(x)dx. 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