Quantitative weighted mixed weak-type inequalities for classical operators

Quantitative Weighted Mixed Weak-Type Inequalities for Classical Operators SHELDY OMBROSI, CARLOS P´ EREZ &JORGELINA RECCHI ABSTRACT.We improve on several mixed weak-type inequalities both for the Hardy-Littlewood maximal function and for Calder´on-Zygmund operators. These types of inequalities were considered by Muckenhoupt and Wheeden and later on by Sawyer estimating the L1,∞(uv) norm of v−1T(fv) for special cases. The emphasis is made in proving new and more precise quantitative estimates involving the Apor A∞constants of the weights involved. 1. INTRODUCTION AND STATEMENTS OF THE MAIN RESULTS Let Mdenote the usual Hardy-Littlewood maximal function; then, according to a fundamental result of B. Muckenhoupt [Mu], Mis a bounded operator on the Lebesgue space Lp(dµ), 1 < p < +∞, if and only if dµ=w(x)dxand the weight wsatisfies the simple geometric condition [w]Ap:=sup Q1 |Q|ZQw 1 |Q|ZQw1−p′p−1 <∞, where the supremum is taken over all cubes Qin Rn. This is the celebrated Muckenhoupt Apcondition. A similar result holds in the case p=1: namely, M is of weak type (1,1)with respect to µ; that is, M:L1(µ) →L1,∞(µ) if and only if dµ=w(x) dxand the weight wsatisfies the A1condition, [w]A1:=sup Q1 |Q|ZQw(essinfQw)−1<∞, where, again, the supremum is taken over all cubes Qin Rn. 615 Indiana University Mathematics Journal c , Vol. 65, No. 2 (2016) 616 SHELDY OMBROSI, CARLOS P´ EREZ &JORGELINA RECCHI Since the Aptheorem of Muckenhoupt plays a central role in modern harmonic analysis, different proofs besides the original in [Mu] have been considered in the literature. In particular, E. Sawyer tried in [Sa] the following approach based on the factorization theorem for Apweights of P. Jones (see [GCRdF]). Recall that a weight wsatisfies the Apcondition if and only if there are two A1weights u and vsuch that w=uv1−p. Then, if the operator Sf =M(vf ) vis defined, the boundedness of Mon Lp(w) may be rewritten as (1.1) ZRn|Sf|puv dx≤cZRn|f|puv dx. Observe now that since v∈A1,Mv ≤[v]A1v, and hence Sis bounded in L∞(uv). Therefore, if we show that Sis of weak type (1,1)with respect to the measure uv dx, we can apply the Marcinkiewicz interpolation theorem to derive (1.1). This is precisely the statement of the following theorem from [Sa]. Theorem 1.1. If u,v∈A1(R), then    M(g) v   L1,∞(uv) ≤ckgkL1(u), where cdepends only on the A1constant of uand the A1constant of v. This shows that the operator Sf =v−1M(vf ) is of weak type (1,1)with respect to the measure vu dx. In the same article, Sawyer conjectured that this theorem should also hold for the maximal function in Rnand for the the Hilbert transform Hinstead of M. The article of Sawyer was also very much motivated by a previous work of B. Muckenhoupt and R. Wheeden [MW]. This time, the main result of this paper holds for both the one-dimensional Hardy-Littlewood maximal function and the Hilbert transform. To be more precise, the main result proved in [MW] is the following. Theorem 1.2. Let w∈A1(R); there exists then a constant csuch that kM(fw−1)wkL1,∞(R)≤ckfkL1(R) and kH(f w−1)wkL1,∞(R)≤ckfkL1(R). In [C-UMP1], the authors extended both Theorems 1.1 and 1.2 to Rn, including in particular the conjectures formulated by Sawyer mentioned above. The precise result is the following. Quantitative Weighted Mixed Weak-Type Inequalities for Classical Operators 617 Theorem 1.3. Suppose u∈A1and that either v∈A1or v∈A∞(u); then, there exists a constant csuch that    M(fv) v   L1,∞(uv) ≤ckfkL1(uv) (1.2) and    T(fv) v   L1,∞(uv) ≤ckfkL1(uv), where Mis the Hardy-Littlewood maximal operator and Tis a Calder´ on-Zygmund operator. We note that this result holds for T∗, the maximal singular integral operator, instead of T. Given weights uand v, by v∈A∞(u), we mean that vsatisfies the A∞condition defined with respect to the measure udx(as opposed to Lebesgue measure). A more precise definition is given in Section 2below. We emphasize that this theorem contains both Theorems 1.1 and 1.2 as particular cases. Indeed, the case of the first theorem is clear. For the second, if w∈A1, we let u=wand v=w−1. Then, uv =1∈A∞, and thus v∈A∞(u) by Lemma 2.1 and Observation 2.2. To prove Theorem 1.3, the authors show that it suffices to prove the result for the dyadic maximal function Mdby proving an extrapolation-type theorem, Theorem 1.5 below, that allows one to replace Tor Mby Md. To be more precise, the combination of the following two theorems from [C-UMP1] proves Theorem 1.3. Theorem 1.4. Suppose that u∈A1and that either v∈A1or v∈A∞(u); then, there exists a constant csuch that (1.3)    Md(fv) v   L1,∞(uv) ≤ckfkL1(uv). Theorem 1.5. Given a family Fof a pair of functions, suppose that for some p∈(0,∞)and for every w∈A∞, kfkLp(w) ≤CkgkLp(w), for all (f , g) ∈ F such that the left-hand side is finite, and where Cdepends only on the A∞constant of w. Then, for all weights u∈A1and v∈A∞, kfv−1kL1,∞(uv) ≤Ckgv−1kL1,∞(uv) (f , g) ∈ F. Here, Fdenotes a family of ordered pairs of non-negative, measurable functions (f , g). Theorem 1.5 from [C-UMP1] is used to pass from Mto Md, since by standard methods, for every p∈(0,∞)and every w∈A∞, kM(fv)kLp(w) ≤ckMd(f v)kLp(w), 618 SHELDY OMBROSI, CARLOS P´ EREZ &JORGELINA RECCHI where the constant cinvolves the A∞constant of w. However, there are recent results showing that Theorem 1.5 can be avoided in the transition from Mto Md. Indeed, using for instance [HP, p. 792], we have that Mf ≤cnX α∈{0,1/3}n Mα df, where Mα dis an appropriate shifted dyadic maximal function with similar properties as Md. Thus, the expression on the left in (1.2) is bounded by a dimensionalconstant multiple of the corresponding expression for Mα d. Since each of these Mα d has similar properties as Md, the corresponding proof of (1.3) is exactly the same. In [C-UMP1], the authors conjectured that Theorem 1.4 still holds under milder hypotheses on the weight v. To be more precise, the authors state what is now known as “Sawyer’s conjecture,” although E. Sawyer never asserted it. The conjecture is the following. Conjecture 1.6. Suppose that u∈A1and v∈A∞. Then, there exists a constant csuch that    Md(fv) v   L1,∞(uv) ≤ckfkL1(uv). Note that if v∈A∞(u) (we always assume u∈A1), then v∈A∞(see Lemmas 2.1 and 2.3). This conjecture has been open for several years and has been studied by different authors. In this paper, we try to understand the difficulties of this conjecture, and propose alternative ways to prove it. We will also study how the constants of the weights uand vare reflected in these inequalities; that is, we look for quantitative versions of this type of inequality. The first question we pose concerning Sawyer’s theorem is the following: What is the sharp dependence on the constants of the weights u and vwhen both are in A1? Following the proof given in [C-UMP1], which is an adaptation of the original proof given by Sawyer in [Sa] for the real line, we show the dependence on the weight constants. More specifically, we prove the following result. Theorem 1.7. If u∈A1and v∈A1, there exists a dimensional constant csuch that (1.4)    Md(fv) v   L1,∞(uv) ≤cu2 A1v4 A1kfkL1(uv). The proof may be found in Section 7. We believe that the dependency on the constants in inequality (1.4) is not sharp, since the method does not seem to be adequate. Trying to understand this issue, we will focus on the special case u=1 that is interesting in its own right. The finiteness of the estimate in this special case is assured by Theorem 1.3 by Quantitative Weighted Mixed Weak-Type Inequalities for Classical Operators 619 assuming even a weaker condition on vthan A1, namely, v∈A∞(u) =A∞. The method we use is different from the one considered in the proof of Theorem 1.7 allowing us to obtain more precise estimates. In particular, we will prove the linearity of the constant bound of the weight vif we assume the stronger condition v∈A1and the result is sharp. Our theorem is the following. Theorem 1.8. Let v∈A1. There exists a dimensional constant c, independent from [v]A1, such that (1.5)    M(f) v   L1,∞(v) ≤c[v]A1kfkL1(Rn). Furthermore, the linear dependence on [v]A1is sharp. However, we want to understand the more general case. Problem 1.9. Find an increasing function ϕ:[1,∞]→[1,∞]for which the following inequality holds whenever v∈A∞: (1.6)    M(f) v   L1,∞(v) ≤cϕ([v]A∞)kfkL1(Rn), where cis a constant that depends on the dimension. This problem is a special case of Conjecture 1.6 with u=1, and it will be studied in Section 4. The best constant in (1.6), ϕ([v]A∞), is finite by Theorem 1.3. Our goal is to determine the best dependence on the constant of the weight v, or, in other words, to find the smallest function ϕ. Recall that A∞=Sp≥1Ap, and that, if w∈A∞we use the weight constant (1.7) [w]A∞:=sup Q 1 w(Q) ZQM(χQw) dx, called the Fujii-Wilson constant in some recent papers. We could use instead the constant defined by Hruˇsˇcev in [Hr], which is more natural; however, it was shown in [HP] that it is much larger than the one given by the functional (1.7). We note here that a condition on the weight vin (1.5) or (1.6) must be taken into account. Indeed, there are estimates like (1.8)    M(f) Mw    L1,∞(Mw) ≤ckfkL1(Rn), namely with v=Mw, that are false for a general function wor measure. This will be shown in Section 5where, furthermore, an interesting relationship with the two weight problem for singular integrals is implicit in the argument. In general, weights of the form Mw are not A∞weights but small perturbations, namely, when v=(Mw)δ,δ∈(0,1), makes the inequality to be true, since 620 SHELDY OMBROSI, CARLOS P´ EREZ &JORGELINA RECCHI in this case v∈A1and Theorem 1.5 applies. It is interesting that in special situations and for large perturbations of the weight, the result is still true. Indeed, if v(x) = |x|−nr ≈(Mδ)rwith r > 1, then there is a finite constant csuch that    M(f) v   L1,∞(v) ≤ckfkL1(Rn), with the result being false in the case r=1. This was proved in dimension one by Andersen and Muckenhoupt in [AM], and by Mart´ ın-Reyes, Ortega Salvador, and Sarri´on Gavi´an [MOS] in higher dimensions. We note that these weights v(x) = |x|−nr are not A∞weights. In view of Theorem 1.8 and the case v=1, we state the following conjecture for the general case. Conjecture 1.10. Let u∈A1and v∈A1; then, there exists a dimensional constant csuch that    Md(fv) v   L1,∞(uv) ≤c[u]A1[v]A1kfkL1(uv). To see that the dependency cannot be better than [u]A1[v]A1, we prove the following result that strengthens our conjecture. Theorem 1.11. Let u∈A1,v∈A1. If    Md(fv) v   L1,∞(uv) ≤cϕ([u]A1, [v]A1)kfkL1(uv), then there is a constant cindependent of the weights such that ϕ([u]A1, [v]A1)≥c[u]A1[v]A1. Another related problem, partly intermediate between the previous two problems, would be to determine how the dependence on the constant [v]Apis if we assume that v∈Apfor some p≥1. We should also take into account that Theorem 1.8 gives the sharp dependence on the real line when assuming the stronger assumption v∈A1. Based on this, we state the following conjecture. Conjecture 1.12. Let v∈Ap,p≥1; then, there exists a dimensional constant csuch that    Md(fv) v   L1,∞(v) ≤c[v]ApkfkL1(v). We were not able to prove this conjecture, but we have obtained the following result using an adequate Calder´on-Zygmund decomposition that involves the A∞ constant of the weight. Quantitative Weighted Mixed Weak-Type Inequalities for Classical Operators 621 Theorem 1.13. Let v∈Ap,p≥1; then, there exists a dimensional constant c such that    Md(fv) v   L1,∞(v) ≤c[v]A∞max{p, log(e +[v]Ap)}kfkL1(v). Corollary 1.14. Let v∈Ap,p≥1; then, there exists a dimensional constant c such that    Md(fv) v   L1,∞(v) ≤Cn[v]Apmax{p, log(e +[v]Ap)}kfkL1(v). We also try to improve the dependency on the weight constant using some other refined constants that were introduced in [HP] and formalized in the work of Lerner and Moen [LM]. Theorem 1.15. Let v∈Ap,p≥1; then, there exists a dimensional constant c such that    Md(fv) v   L1,∞(v) ≤cp[v](Ap)1/p(Aexp ∞)1/p′log(e +[v](Ap)1/p(Aexp ∞)1/p′)kfkL1(v). We defer to Section 2for the definition of [v](Ap)1/p(Aexp ∞)1/p′. In this paper, we will also study similar problems for Calder´on-Zygmund operators instead of the Hardy-Littlewood maximal function. In particular, we improve the following theorem from [HP]. Theorem 1.16. Suppose that Tis a Calder´ on-Zygmund operator; then, there is a dimensional constant c such that, for any v∈A1,    T(fv) v   L1,∞(v) ≤c[v]A1log(e +[v]A∞)kfkL1(v). This theorem improved the following result previously obtained in [LOP2]:    T(fv) v   L1,∞(v) ≤c[v]A1log(e +[v]A1)kfkL1(v). In Section 5, we give a version of Corollary 1.14 for Calder´on-Zygmund operators. We prove the following result. Theorem 1.17. Suppose Tis a Calder´ on-Zygmund Operator; then, there is a dimensional constant c such that, for any v∈Ap,    T(fv) v   L1,∞(v) ≤c[v]Apmax{p, log(e +[v]Ap)}kfkL1(v). 622 SHELDY OMBROSI, CARLOS P´ EREZ &JORGELINA RECCHI 2. PRELIMINARIES As usual, a weight will be a non-negative locally integrable function. Given a weight w,p∈(1,∞), and a cube Q, we denote Ap(w;Q) :=1 |Q|ZQw 1 |Q|ZQw1−p′p−1 =w(Q)σ(Q)p−1 |Q|p, where σ=w−1/(p−1). When p=1 we define the limiting quantity as A1(w;Q) :=1 |Q|ZQw(inf Qw)−1=lim p→1Ap(w, Q). For p= ∞, we consider two constants. The first constant is defined as a limit of the Ap(w;Q) constants Aexp ∞(w;Q) :=1 |Q|ZQwexp1 |Q|ZQlogw−1=lim p→∞ Ap(w, Q). To define the second constant, we let AW ∞(w;Q) :=1 w(Q) ZQM(χQw), and define [w]Ap=sup Q Ap(w;Q), kwkA∞=sup Q Aexp ∞(w;Q), [w]A∞=sup Q AW ∞(w;Q). We write w∈Apif [w]Ap<∞and w∈A∞if kwkA∞<∞or [w]A∞<∞. The constant kwkA∞was defined by Hruˇsˇcev in [Hr]. The constant [w]A∞was defined by Fujii in [F] and rediscovered by M. Wilson in [W1,W3], who also showed that both constants define the class A∞. In [HP], the authors proved the estimate [w]A∞≤cnkwkA∞, and provided examples showing that kwkA∞can be exponentially larger than [w]A∞. We now define the mixed-type constants. Given 1 ≤p < ∞and α,β≥0, motivated by some results for the two weighted estimates for the maximal function in [HP], Lerner and Moen in [LM] defined the mixed constants [w](Ap)α(Ar)β=sup Q Ap(w;Q)αAr(w;Q)β,1≤r < ∞, the exponential mixed constants (2.1) [w](Ap)α(Aexp ∞)β=sup Q Ap(w;Q)αAexp ∞(w;Q)β, Quantitative Weighted Mixed Weak-Type Inequalities for Classical Operators 623 and the Fujii-Wilson mixed constants [w](Ap)α(AW ∞)β=sup Q Ap(w;Q)αAW ∞(w;Q)β. If α > 0, the class of weights that satisfies [w](Ap)α(AW ∞)β<∞is simply the class Ap, since maxwα Ap,wβ A∞≤[w](Ap)α(AW ∞)β≤wα+β Ap. Analogously, a weight wsatisfies [w](Ap)α(Aexp ∞)β<∞if and only if wis in Ap such that the inequality holds for the exponential mixed constant. In [LM], the authors show that if 0 < α ≤β≤1 and w∈Ap, then (2.2) [w](Ap)α(Aexp ∞)1−α≤[w](Ap)β(Aexp ∞)1−β. We finish this section by defining the generalized A∞class of weights A∞(µ) where µis a doubling measure. To do this, we recall some well-known definitions about generalized Hardy-Littlewood maximal operators. For a complete account, we refer the reader to [D,GCRdF]. Given a doubling measure µ, we define the maximal operator Mµby Mµf(x) =sup Q∋x 1 µ(Q) ZQ|f(y)|dµ(y). For 1 < p < ∞, given a weight w, we say that w∈Ap(µ) if, for all cubes Q, 1 µ(Q) ZQw(x) dµ(x) 1 µ(Q) ZQw(x)1−p′dµ(x)p−1 ≤C. We say that w∈A1(µ) if Mµw(x) ≤Cw(x). We denote the union of all the Ap(µ) classes by A∞(µ), that is, A∞(µ) =[ p≥1 Ap(µ). Since µis doubling, then Mµis bounded on Lp(w dµ), 1 < p < ∞, if and only if w∈Ap(µ). As usual, when µis the Lebesgue measure we omit the subscript µ, and write simply Mor Ap. Also, if µis absolutely continuous given by the weight u, we then simply write Ap(u), 1 ≤p≤ ∞. The next two lemmas were proved in [C-UMP2]. Lemma 2.1. If u∈A1and v∈A∞(u), then uv ∈A∞. In particular, if v∈Ap(u),1≤p < ∞, then uv ∈Ap. Observation 2.2. If u∈A1, then v∈A∞(u) if and only if uv ∈A∞. Lemma 2.3. If u∈A1and uv ∈A∞, then v∈A∞. 630 SHELDY OMBROSI, CARLOS P´ EREZ &JORGELINA RECCHI 5. COUNTER-EXAMPLES In this section, we show that inequality (1.8) is false. To do this, we proceed by contradiction, assuming that this inequality holds. We begin with the following duality argument for any weight w: kTfkLp(w) =sup h:khkLp′(w)=1   ZRnTfhw dx   . Fixing one of these h, we have ZRnTfhw dx=ZRnf Tt(hw) dx=ZRnfTt(h w) Mw Mw dx and then   ZRnTfhw dx   ≤ kfkLp(Mw)      Tt(hw) Mw     Lp′(Mw) = kfkLp(Mw) kTtfkLp′(Mw)1−p′). We now use the following lemma, which is a particular version of the classical estimate of Coifman-Fefferman for any Calder´on-Zygmund operator T: let p∈(0,∞), and let w∈A∞; then, there is a constant cdepending upon p, T and the A∞constant of w, such that (5.1) kTfkLp(w) ≤cT,p,[w]A∞kMf kLp(w). Then, as a consequence we have the following special situation. Let wbe any weight, and let p∈(1,∞). Then, there is a constant depending only on pand T such that (5.2) kTfkLp(Mw)1−p)≤ckMfkLp(Mw)1−p). This follows from (5.1) and the fact that (Mw)1−p∈A∞. Indeed, since (Mw)1−p=(Mw)δ(1−2p) ∈A2pand δ=p−1 2p−1<1 2, we have [(Mw)1−p]A∞≤[(Mw)δ]2p−1 A1≤cp n. (It should be mentioned that (5.2) was improved in [LOP3] and later in [LOP1]. In these papers, the relevance was the sharpness of the constant cin terms of p, which behaves linearly in p, but is not important in our context. See also [R] for a similar estimate within the fractional integrals context.) Then, since Ttis also a Calder´on-Zygmund operator, we apply (5.2):    ZRnTfhw dx   ≤cp,T kfkLp(Mw) kM(hw)kLp′(Mw)1−p′ =cp,T kfkLp(Mw)    M(hw) Mw    Lp′(Mw) . Quantitative Weighted Mixed Weak-Type Inequalities for Classical Operators 631 We now apply (1.8), which is equivalent to    M(fw) Mw    L1,∞(Mw) ≤ckfkL1(w), and since the operator f→M(fw)/(Mw) is trivially bounded on L∞with constant 1, we apply the Marcinkiewicz’s interpolation theorem to deduce    ZRnTfhw dx   ≤cpkfkLp(Mw) khkLp′(w) =cpkfkLp(Mw). Then, for any Calder´on-Zygmund operator Tand arbitrary weight w, we have produced the estimate kTfkLp(w) ≤cpkfkLp(Mw). However, this inequality is well known to be false for any p∈(1,∞), as was shown by M. Wilson in [W2] for the simplest case, namely, the Hilbert transform. 6. CALDER ´ ON-ZYGMUND INTEGRAL OPERATOR In this section, we show the following inequality:    T(fv) v   L1,∞(v) ≤Cn[v]Apmax{p, log(e +[v]Ap)}ZRn|f(x)|v(x) dx. For the proof of this inequality, the following two results are needed. The first result was proved in [Hy], and the second result can be found in [GCRdF, p. 413]. Theorem 6.1. Let 1< p < ∞,wan Ap-weight and Ta Calder´ on-Zygmund operator; then, kTfkLp(w) ≤cnpp′wmax(1,1/(p−1)) Ap. Lemma 6.2. Let wbe a weight. There is a dimensional constant cdsuch that, for any cube Qand for any function fsupported in a cube Qwith ZQf (x) dx=0, the following inequality holds: ZRn\2Q|Tf(y)|w(y) dy≤cnZQ|f (y)|Mw(y) dy. The structure of the proof of Theorem 1.17 is similar to that of Theorem 1.13. Proof of Theorem 1.17.Without loss of generality, we assume fis bounded and has compact support. Since v∈Ap, then for all r > p, we have v∈Ar, with [v]Ar≤[v]Ap. Fix t > 0. For now, let r > p be arbitrary, as we will assign a specific value to r. Since v∈Ar, in particular, vdxis a doubling weight. Therefore, we can 632 SHELDY OMBROSI, CARLOS P´ EREZ &JORGELINA RECCHI form the Calder´on-Zygmund decomposition of fat height twith respect to the measure vdx. This yields a collection of disjoint dyadic maximal cubes {Qj}, such that for all Qj, t < 1 v(Qj)ZQj f (x)v(x) dx≤v(Q′ j) v(Qj)v(Q′ j)ZQ′ j f (x)v(x) dx≤2nr [v]Art, where as before Q′ jis the ancestor of Qj, and where the last inequality is obtained by using standard properties of the Apweights (see Proposition 9.1.5 in [G]), and by the maximal property of the Qj. Further, if we let Ω:=SjQj, then f (x) ≤t for almost every x∈Rn\Ω. We decompose fas g+b, where g(x) =     1 v(Qj)ZQj f (x)v(x) dxif x∈Qj, f(x) if x∈Rn\Ω, and we let b(x) =Pjbj(x), with bj(x) =f(x) −1 v(Qj)ZQj f (x)v(x) dxχQj(x). If we use these definitions, we have that g(x) ≤2nr [v]Artfor almost every x∈Rn, and ZQj bj(x)v(x) dx=0. Then, since Tis a sublinear operator, we have that vx∈Rn:|T(fv)(x)| v(x) > t ≤vx∈Rn:|T(gv)(x)| v(x) >t 2 +vx∈˜ Ω:|T(bv)(x)| v(x) >t 2 +vx∈Rn\˜ Ω:|T(bv)(x)| v(x) >t 2 =I1+I2+I3. For I1we combine the Chebyshev inequality with Theorem 6.1, bearing in mind that as v∈Ar, we have v1−r′∈Ar′with [v1−r′]Ar′=vr′−1 Ar. Quantitative Weighted Mixed Weak-Type Inequalities for Classical Operators 633 Now, since the exponent of the constant [v]Ar′in Lemma 6.1 is different if p > 2 or p≤2, we have divided the proof into two cases. Case p > 2. In this case, as r > 2, we have r′<2 and max(1,1/(r′−1)) = 1/(r′−1). Thus, I1≤2r′ tr′ZRn|T(gv)(x)|r′v(x)1−r′dx≤cr′ n tr′rr′vr′ ArZRng(x)r′v(x) dx. Then, since g(x) ≤2nr [v]Artand [v]Ar≤[v]Ap, we obtain that I1≤cr′ n trr′v2r′−1 ArZRng(x)v(x) dx≤2r′(1+n) trr′v2r′−1 ApZRng(x)v(x) dx. As r > p > 2, we choose r=1+max{p, log(e +[v]Ap)}; then, 2> r′=1+1 max{p, log(e +[v]Ap)}. For this reason, rr′behaves like max{p, log(e+[v]Ap)}and [v]2r′−1 Aplike [v]Ap: I1≤Cn t[v]Apmax{p, log(e +[v]Ap)} ×ZRn\Ω f (x)v(x) dx+X j1 v(Qj)ZQj f (x)v(x) dxv(Qj) ≤Cn t[v]Apmax{p, log(e +[v]Ap)}ZRnf (x)v(x) dx. Case p≤2. We choose r=1+2log(e +[v]Ap) > 2≥p; thus, r′=1+1 2log(e +[v]Ap)<2, and max(1,1/(r′−1)) =1/(r′−1). We can now proceed analogously to the previous case: I1≤cr′ d trr′v2r′−1 ApZRng(x)v(x) dx, and therefore, I1≤cd tlog(e +[v]Ap)ZRnf(x)v(x) dx. 634 SHELDY OMBROSI, CARLOS P´ EREZ &JORGELINA RECCHI The estimate for I2follows immediately from the properties of the cubes Qj and from the inequality v(2Q) ≤2np[v]Apv(Q), as follows: I2≤v( ˜ Ω)≤X j v(2Qj)≤2np[v]ApX j 1 tZQj f (x)v(x) dx ≤2np[v]Ap 1 tZRnf (x)v(x) dx. Finally, to be able to estimate I3, we use Lemma 6.2 with w≡1: I3≤2 tZRn\˜ Ω |T(bv)(x)|dx≤2 tX jZRn\2Qj |T(bjv)(x)|dx ≤2 tX jZQj |bj(x)|v(x) dx. If we have used the definition of bj, we have that I3≤cd tkfvkL1(Rn). ❐ 7. ANADAPTATION OF SAWYER’SPROOF WITH CONTROL OF THE CONSTANT In this appendix, we prove Theorem 1.7 using a method similar to the one considered in [C-UMP1] for the proof of Theorem 1.4. The statement of the theorem assumes that the weights belong to the A1class of weights. These weights satisfy a reverse H¨older inequality: namely, if w∈A1, then there are two constants r, c > 1 such that 1 |Q|ZQwr!1/r ≤c |Q|ZQw. However, we note that in the classical proofs there is a bad dependence on the constant c=c(r, [w]A1), and we need a more precise estimate to get our results. Lemma 7.1. Let w∈A1, and let rw=1+1/(2n+1[w]A1). Then, for any cube Q, 1 |Q|ZQwrw!1/rw ≤2 |Q|ZQw. As a consequence, we have that for any cube Qand for any measurable set E⊂Q, w(E) w(Q) ≤2|E| |I|εw , where εw=1/(1+2n+1[w]A1). Quantitative Weighted Mixed Weak-Type Inequalities for Classical Operators 635 The proof of this reverse H¨older inequality can be found in [LOP1], and the consequence is an application of H¨older’s inequality. Proof of Theorem 1.7.Fix t > 0, and define g=fv/t. Then, it is sufficient to show that (7.1) uv x∈Rn|Md(g)(x) > v(x)≤CZRn|g(x)|u(x) dx, for any function gbounded with compact support. Fix a > 2n. For each k∈Z, let {Ik j}be the collection of maximal, disjoint dyadic cubes whose union is the set Ωk= {x∈Rn|Mdv(x) > ak} ∩ {x∈Rn|Mdg(x) > ak}. This decomposition exists since gis bounded and has compact support, so the second set is contained in the union of maximal dyadic cubes. Define Γ= {(k, j) :|Ik j∩ {x:v(x) ≤ak+1}| >0}. As v∈A1, we have Mv(x) ≤[v]A1v(x) almost everywhere. Hence, for (k, j) ∈Γ, ak [v]A1 ≤1 [v]A1 essinfx∈Ik jMdv(x) ≤essinfx∈Ik jv(x)(7.2) ≤1 |Ik j|ZIk j v(x) dx≤[v]A1ak+1. (Intuitively, if (k, j) ∈Γ, then Ik jbehaves like a cube from the Calder´on-Zygmund decomposition of vat height ak.) Then, up to a set of measure zero, we have the following inclusions: for each k, {x∈Rn|ak< v(x) ≤ak+1} ∩ {x∈Rn|Mdg(x) > v(x)} ⊂ [ j:(k,j)∈Γ Ik j. Combining this with (7.2), we get that uv({x∈Rn|Mdg(x) > v(x)})≤a[v]A1X (k,j)∈Γ |Ik j|−1v(Ik j)u(Ik j). Fix N < 0, and define ΓN= {(k, j) ∈Γ|k≥N}. We will show that X (k,j)∈γN |Ik j|−1v(Ik j)u(Ik j)≤CZRn|g(x)|u(x) dx, 636 SHELDY OMBROSI, CARLOS P´ EREZ &JORGELINA RECCHI where the constant Cdoes not depend on N. Inequality (7.1) then follows if we take the limit as N→ −∞. To prove this, we replace the set of cubes {Ik j}by a subset with better properties. First, since v∈A1, we can apply Lemma 7.1; and there exists ε=(1+2n+1[v]A1)−1>0 such that, given any cube Iand E⊂I, (7.3) v(E) v(I) ≤2|E| |I|ε . Fix δsuch that 0 < δ < ε. Define ∆N= {Ik j|(k, j) ∈ΓN}. The cubes in ∆Nare all dyadic, so they are either pairwise disjoint or one is contained in the other. For k > t, since Ωk⊂Ωtand since the cubes Ik jare maximal in Ωk, if It s∩Ik j≠∅, then Ik j⊂It s. In particular, each cube Ik j∈∆Nis contained in SjIN j⊂ {x|Mdg(x) > aN}. As we noted above, the last set is bounded, so ∆N contains a maximal disjoint subcollection of cubes. We form a sequence of sets {Gn}by induction. Let G0be the set of all pairs (k, j) ∈ΓNsuch that Ik jis maximal in ∆N. For n≥0, given the set Gn, define the set Gn+1to be the set of pairs (k, j) ∈ΓNsuch that there exists (t, s) ∈Gnwith Ik j⊊It sand 1 |Ik j|ZIk j u(x) dx > a(k−t)δ 1 |It s|ZIt s u(x) dx,(7.4) 1 |Iℓ i|ZIℓ i u(x) dx≤a(ℓ−t)δ 1 |It s|ZIt s u(x) dx,(7.5) whenever (ℓ, i) ∈ΓNand Ik j⊊Iℓ i⊂It s. Let P=Sn≥0Gn. Given (s, t) ∈P, we refer to the cube It sas a principal cube. Since every cube in ∆Nis contained in a maximal cube, every cube in ∆Nis contained in one or more principal cubes. To continue, we divide the proof into several steps of the same form. We only look at the behavior of the A1-constants, and give the main ideas of the steps. Step 1. We claim that (7.6) X (k,j)∈ΓN |Ik j|−1v(Ik j)u(Ik j)≤CεX (k,j)∈P |Ik j|−1v(Ik j)u(Ik j). To prove this, fix (t, s) ∈P, and let Q=Q(t, s) be the set of indices (k, j) ∈ΓN such that Ik j⊂It sand It sis the smallest principal cube containing Ik j. In particular, each Ik jis not a principal cube unless it equals It s. Thus, by (7.5), and since Ik j⊂ {x|Mdv(x) > ak}, X (k,j)∈Q |Ik j|−1v(Ik j)u(Ik j)≤ |It s|−1u(It s)X k≥t a(k−t)δv(It s∩ {x|Mdv(x) > ak}). Quantitative Weighted Mixed Weak-Type Inequalities for Classical Operators 637 By (7.3) and (7.2), and since v∈A1, v(It s∩ {x|Mdv(x) > ak})≤2v2ε A1aεa(t−k)εv(It s). Combining these inequalities, we see that X (k,j)∈Q |Ik j|−1v(Ik j)u(Ik j)≤Cε|It s|−1u(It s)v(It s), where Cε=(a2ε−δ/(aε−δ−1))2[v]2ε A1. If we now sum over all (s, t) ∈P, we get (7.6), since S(t,s)∈PQ(t, s) =ΓN. Step 2. For each k, let {Jk i}be the collection of maximal disjoint cubes whose union is {x|Mdg(x) > ak}. Then, ak<1 |Jk i|ZJk i g(x) dx. For each j, we have Ik j⊂ {x|Mdg(x) > ak}, so there exists a unique i=i(j, k) such that Ik j⊂Jk i. Hereafter, the index iwill always be this function of (k, j). Hence, by (7.6) and by (7.2), X (k,j)∈ΓN |Ik j|−1v(Ik j)u(Ik j)≤Cεa[v]A1ZRnh(x)g(x) dx, where h(x) =P(k,j)∈P|Jk i|−1χJk i(x)u(Ik j). To complete the proof, we show that for each x,h(x) ≤Cu(x). Fix x∈Rn; without loss of generality, we may assume that u(x) is finite. For each k, there exists at most one cube Jk bsuch that x∈Jk b. If it exists, denote this cube by Jk. Define Pk= {(k, j) ∈P|Ik j⊂Jk}, and G= {k|Pk≠∅}. We form a sequence {km}by induction. If k∈G, then k≥N, so let k0be the least integer in G. Given Km,m≥0, choose km+1> kmin Gsuch that 1 |Jkm+1|ZJkm+1 u(y) dy > 2 |Jkm|ZJkm u(y) dy,(7.7) 1 |Jℓ|ZJℓu(y) dy≤2 |Jkm|ZJkm u(y) dy,(7.8) where km≤ℓ < km+1, ℓ ∈G. Since u(x) is finite, the sequence {Km}only contains a finite number of terms. Then, by (7.8), we have h(x) ≤X m 2 |Jkm|ZJkm u(y) dyX ℓ∈G, km≤ℓ<km+1X (ℓ,j)∈Pℓ u(Iℓ j) u(Jℓ). 638 SHELDY OMBROSI, CARLOS P´ EREZ &JORGELINA RECCHI We claim that (7.9) X ℓ∈G, km≤ℓ<km+1X (ℓ,j)∈Pℓ u(Iℓ j) u(Jℓ)≤C9. Given this, we are done: since the sequence {km}is finite, let mbe the largest index. Then, by (7.7) and (7.9), h(x) ≤2C9 2−1 2m![u]A1u(x). Therefore, to complete the proof we must show (7.9). We do this in two steps. Step 3. The authors of [C-UMP1] proved that if (ℓ, j) ∈Pℓ,km≤ℓ < km+1, then (7.10) 1 |Iℓ j|ZIℓ j u(y) dy > a(ℓ−km)δ 2[u]A1 1 |Jℓ|ZJℓu(y) dy. Step 4. We now prove (7.9). By (7.10), and again since u∈A1, if y∈Iℓ j, then λ=a(ℓ−km)δ 2[u]A1 u(Jℓ) |Jℓ| 1 [u]A1 < u(y); hence, [ j:(ℓ,j)∈Pℓ Iℓ j⊂ {x∈Jℓ|u(x) > λ}. For ℓfixed, the cubes Iℓ jare disjoint. Therefore, since we have u∈A1, there exist ν=(1+2n+1[u]A1)−1such that X j:(ℓ,j)∈Pℓ u(Iℓ j)≤21+νu(Jℓ)u2ν A1a(km−ℓ)δν . Therefore, we have that X ℓ∈G, km≤ℓ<km+1X (ℓ,j)∈Pℓ u(Iℓ j) u(Jℓ)≤C9, where C9=21+ν[u]2ν A1aδν/(aδν −1). Then, the constant Cof Theorem 1.7 behaves like aε−δ aε−δ−123+νaε+2v2ε+2 A1u2ν+1 A1 aδν aδν −1 2−1 2m!≈v4 A1u2 A1. ❐ Quantitative Weighted Mixed Weak-Type Inequalities for Classical Operators 639 Acknowledgements. The second author is supported by the Spanish government grant MTM-2014-53850-P, and the first and third authors are supported by Universidad Nacional del Sur and CONICET. The authors would also like to thank the referee, whose suggestions and comments have been very helpful in improving the presentation of the paper. REFERENCES [AM] K. F. ANDERSEN AND B. MUCKENHOUPT,Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), no. 1, 9–26. MR665888 (83k:42018). [Bu] S. M. BUCKLEY,Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. 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