Sharp weighted bounds involving A∞

arXiv:1103.5562v1 [math.CA] 29 Mar 2011 SHARP WEIGHTED BOUNDS INVOLVING A∞ TUOMAS HYTÖNEN AND CARLOS PÉREZ Abstract. We improve on several weighted inequalities of recent interest by replacing a part of the Apbounds by weaker A∞estimates involving Wilson’s A∞constant [w]′ A∞:= sup Q 1 w(Q)ˆQ M(wχQ). In particular, we show the following improvement of the first author’s A2 theorem for Calderón–Zygmund operators T: kTkB(L2(w)) ≤cT[w]1/2 A2[w]′ A∞+ [w−1]′ A∞1/2. Corresponding Aptype results are obtained from a new extrapolation theorem with appropriate mixed Ap–A∞bounds. This uses new two-weight estimates for the maximal function, which improve on Buckley’s classical bound. We also derive mixed A1–A∞type results of Lerner, Ombrosi and the second author (Math. Res. Lett. 2009) of the form: kTkB(Lp(w)) ≤c pp′[w]1/p A1([w]′ A∞)1/p′,1< p < ∞, kT fkL1,∞(w)≤c[w]A1log(e+ [w]′ A∞)kfkL1(w). An estimate dual to the last one is also found, as well as new bounds for commutators of singular integrals. 1. Introduction and statements of the main results The weights wfor which the usual operators Tof Classical Analysis (like the Hardy–Littlewood maximal operator, the Hilbert transform, and general classes of Calderón–Zygmund operators) act boundedly on Lp(w)were identified in the 1970’s in the works of Muckenhoupt, Hunt, Wheeden, Coifman and Fefferman [5, 15, 31]. This class consists of the Muckenhoupt Apweights, defined by the condition that (see [13]) [w]Ap:= sup Q− ˆQ w− ˆQ w−1/(p−1)p−1<∞, p ∈(1,∞), where the supremum is over all cubes in Rd. Hence it is shown for any of these important operators T, whether it is linear or not, that kTkB(Lp(w)) := sup f6=0 kT fkLp(w) kfkLp(w) is finite if and only if [w]Ap<∞. 2010 Mathematics Subject Classification. 42B25 (Primary); 42B20, 42B35 (Secondary). Key words and phrases. Weighted norm inequalities, Apweights, sharp estimates, maximal function, Calderón–Zygmund operators. T. H. was supported by the Academy of Finland, projects 130166, 133264 and 218148. C. P. was supported by the Spanish Ministry of Science and Innovation grant MTM2009-08934 and by the Junta de Andalucía, grant FQM-4745. 1 2 T. HYTÖNEN AND C. PÉREZ 1.A. The Aptheory. It is a natural question to look for optimal quantitative bounds of kTkB(Lp(w)) in terms of [w]Ap. The first author who studied that question was S. Buckley in his 1992 Ph.D. thesis (see [3]) who proved kMkB(Lp(w)) ≤cp,d [w] 1 p−1 Ap1< p < ∞,(1.1) where Mis the usual Hardy-Littlewood maximal function on Rd. However, there has been a big impetus on finding such precise dependence for more singular operators after the work of Astala, Iwaniec and Saksman [1] due to the connections with sharp regularity results for solutions to the Beltrami equation. The key fact was to prove that the operator norm of the Beurling-Ahlfors transform on L2(w) grows linearly in terms of the A2constant of w. This was proved by S. Petermichl and A. Volberg [43] and by Petermichl [41, 42] for the Hilbert transform and the Riesz transforms. To be precise, in these papers it has been shown that if Tis any of these operators, then kTkB(Lp(w)) ≤cp,T [w]max{1,1 p−1} Ap.(1.2) The exponents are optimal in the sense that the exponent cannot be replaced by any smaller quantity. It was conjectured then that the same estimate holds for any Calderón-Zygmund operator T. This was first proven for special classes of integral transforms [7, 23] and finally, for general Calderón–Zygmund operators by the first author in [16], using the main result from [37] where it is shown that a weak type estimate is enough to prove the strong type. A direct proof of this result can be found in [20]. Other related work are [8, 19, 22, 24, 47]. The main purpose of this paper is to show that these results can be further improved. To do this, we recall the following definitions of the A∞constant of a weight w: First, there is the notion introduced by Hruščev [14] (see also [13]): [w]A∞:= sup Q− ˆQ wexp − ˆQ log w−1, and second, the smaller (as it turns out) quantity, which appeared with a different notation in the work of Wilson [48, 49, 50] and was recently termed the ‘A∞ constant’ by Lerner [24, Section 5.5]: [w]′ A∞:= sup Q 1 w(Q)ˆQ M(wχQ). Observe that cd[w]′ A∞≤[w]A∞≤[w]Ap,∀p∈[1,∞), where the second estimate is elementary, and the first will be checked in Proposition 2.2. While the constant [w]A∞is more widely used in the literature, and also more flexible for our purposes, it is of interest to observe situations, where the smaller constant [w]′ A∞is sufficient for our estimates, thereby giving a sharper bound. Now, if σ=w−1/(p−1) is the dual weight of w, we also have [σ]p−1 A∞≤[σ]p−1 Ap′= [w]Ap. The point here is that these quantities can be much smaller for some classes SHARP WEIGHTED BOUNDS INVOLVING A∞3 of weights. Our results will be of the form kTkB(Lp(w)) ≤cp,T X[w]α(p) Ap[w]β(p) A∞[σ](p−1)γ(p) A∞, sometimes even with the smaller [ ]′ A∞constant instead of [ ]A∞, where the sum is over at most two triplets (α, β, γ), and the exponents satisfy α(p) + β(p) + γ(p) = τ(p), where τ(p)is the exponent from the earlier sharp results, as it should. However, we will have α(p)< τ(p), which shows that part of the necessary Ap control may in fact be replaced by weaker A∞control. As in the usual case, a key point to understand our main result for Calderón– Zygmund operators is to consider first the case p= 2. 1.3. Theorem. Let Tbe a Calderón–Zygmund operator, w∈A2and σ=w−1. Then there is a constant c=cd,T such that kTkB(L2(w)) ≤c[w]1/2 A2[w]′ A∞+ [w−1]′ A∞1/2 ≤c[w]1/2 A2[w]A∞+ [w−1]A∞1/2. (1.4) We will prove this by following the approach form [16, 20] to the A2theorem kTkB(L2(w)) ≤cT[w]A2, and modifying the proof at some critical points. Indeed, the original argument uses the A2property basically twice, each producing the factor [w]1/2 A2, and it suffices to observe that only the A∞property is actually needed in one of these estimates. An interesting consequence of this theorem is the following: for any fixed Calderón– Zygmund operator T, we have inf w∈A2 kTkB(L2(w)) [w]A2 = 0.(1.5) This follows once we describe, in Section 8, a family of weights w∈A2for which both [w]′ A∞and [σ]′ A∞(and even [w]A∞and [σ]A∞) grow slower than [w]A2. In particular, the “reverse A2conjecture” [w]A2≤cTkTkB(L2(w)) is false. The result for p6= 2 will be obtained by means of a new quantitative variant of the Rubio de Francia extrapolation theorem adapted to the A∞control, which we discuss further below. 1.6. Corollary. Let Tbe a Calderón–Zygmund operator and let p∈(1,∞). Then if w∈Apand σ=w−1/(p−1), we have kTkB(Lp(w)) .[w]2/p−1/2 Ap[w]1/2 A∞+ [σ](p−1)/2 A∞([σ]′ A∞)2/p−1 .[w]2/p Ap([σ]′ A∞)2/p−1for p∈(1,2], (1.7) and kTkB(Lp(w)) .[w]2/p−1/[2(p−1)] Ap[w]1/[2(p−1)] A∞+ [σ]1/2 A∞([w]′ A∞)1−2/p .[w]2/p Ap([w]′ A∞)1−2/p for p∈[2,∞). (1.8) Here the simpler forms of the estimates in (1.7) and (1.8) are almost as good as the more complicated ones, since for many common weights, like power weights, we have [w]A∞+ [σ]p−1 A∞ h[w]Ap; see Section 8. These two statements (1.7) and (1.8) are actually equivalent to each other by using kTkB(Lp(w)) =kT∗kB(Lp′(σ)) 4 T. HYTÖNEN AND C. PÉREZ and the fact that T∗is also a Calderón–Zygmund operator. 1.B. The maximal function and extrapolation with A∞control. As mentioned above, a key point to understand Corollary 1.6 is to prove a version of the quantitative extrapolation theorem adapted to A∞control. The proof of this theorem requires to study the corresponding question for the Hardy– Littlewood maximal function, which we first do in a two-weight setting. We need a new two-weight constant Bp[w, σ]defined by the functional Bp[w, σ] := sup Q− ˆQ w− ˆQ σpexp − ˆQ log σ−1.(1.9) which clearly satisfies [w]Ap≤Bp[w, σ]≤[w]Ap[σ]A∞. 1.10. Theorem. Let Mbe the Hardy-Littlewood maximal operator and let p∈ (1,∞). Then we have the estimates kM(fσ)kLp(w)≤Cd·p′·Bp[w, σ]1/pkfkLp(σ),(1.11) and kM(fσ)kLp(w)≤Cd·p′·[w]Ap[σ]′ A∞1/pkfkLp(σ).(1.12) We refer to Section 4 for the proof and for more information and background about this two-weight estimate for M. By a well-known change-of-weight argument, (1.12) implies: 1.13. Corollary. For Mand pas above, and σ=w−1/(p−1), we have kMkB(Lp(w)) ≤Cd·p′·[w]Ap[σ]′ A∞1/p.(1.14) This improves on Buckley’s theorem kMkB(Lp(w)) ≤Cd·p′·[w]1/(p−1) Ap. Corollary 1.13, at least for p= 2, was also independently discovered by A. Lerner and S. Ombrosi [25]. We now recall the following quantitative version of Rubio de Francia’s classical extrapolation theorem due to Dragičević, Grafakos, Pereyra, and Petermichl [10]: If an operator Tsatisfies kTkB(Lr(w)) ≤ϕ([w]Ar) for a fixed increasing function ϕand for all w∈Ar, then it satisfies a similar estimate for all p∈(1,∞): kTkB(Lp(w)) ≤2ϕcp,r,d[w]max{1,(r−1)/(p−1)} Ap; in particular, kTkB(Lr(w)) .[w]τ(r) Arimplies that kTkB(Lp(w)) .[w]τ(r) max{1,(r−1)/(p−1)} Ap. With our new quantitative estimates involving both Apand A∞control, it seems of interest to extrapolate such bounds as well. Hence we consider weighted estimates of the form kT fkLr(w)≤ϕ[w]Ar,[w]A∞,[w−1/(r−1)](r−1) A∞kfkLr(w)(1.15) SHARP WEIGHTED BOUNDS INVOLVING A∞5 where ϕ: [1,∞)3→[0,∞)is an increasing function with respect to each of the variables. An example is our bound for singular integrals (1.3), where ϕ(x, y, z) = Cx1/2(y+z)1/2.(1.16) We now aim to extrapolate bounds like (1.15) from the given r∈(1,∞)to other exponents p∈(1,∞). 1.17. Theorem (Lower Extrapolation).Suppose that for some rand every w∈Ar, an operator Tsatisfies (1.15). Then for every p∈(1, r), it satisfies kT fkLp(w)≤2ϕ(2kMkB(Lp(w)))r−p[w]Ar,[w]A∞,[w−1/(p−1)](p−1) A∞kfkLp(w) ≤2ϕ(cd([w]Ap[w−1/(p−1)]′ A∞)1/p)r−p ×[w]Ap,[w]A∞,[w−1/(p−1)](p−1) A∞kfkLp(w) In typical applications, like (1.16), the function ϕwill have a homogeneity of the form ϕ(λx, λy, λz) = λsϕ(x, y, z), and hence the common factor (2kMkB(Lp(w)))r−p≤(cd([w]Ap[w−1/(p−1)]′ A∞)1/p)r−p may be extracted out of ϕ. Observe that the condition (1.15) is of course implied by the stronger inequality kT fkLr(w)≤ϕ[w]Ar, c−1 d[w]′ A∞,(c−1 d[w−1/(r−1)]′ A∞)(r−1)kfkLr(w); however, even if we have this stronger inequality to start with (as is the case with the A2theorem for Calderón–Zygmund operators), we do not know how to exploit it to get a stronger conclusion than what we can derive from (1.15). A related difficulty will be pointed out in the proof. This is why we restrict to the assumption (1.15) only. 1.18. Theorem (Upper Extrapolation).Suppose that for some rand every w∈Ar, an operator Tsatisfies (1.15). Then for every p∈(r, ∞), it satisfies kT fkLp(w)≤2ϕ(2kMkB(Lp′(w1−p′)))(p−r)/(p−1) ×[w](r−1)/(p−1) Ap,[w](r−1)/(p−1) A∞,[w−1/(p−1)](r−1) A∞kfkLp(w) ≤2ϕcd[w]1/p Ap([w]′ A∞)1/p′(p−r)/(p−1) ×[w](r−1)/(p−1) Ap,[w](r−1)/(p−1) A∞,[w−1/(p−1)](r−1) A∞kfkLp(w). It is immediate to check that Theorems 1.17 and 1.18, in combination with Theorem 1.3, give Corollary 1.6. In fact, thanks to the mentioned equivalence of the two parts (1.7) and (1.8) of Corollary 1.6, we would only need one of Theorems 1.17 and 1.18 to deduce this corollary. But for other classes of operators without a self-dual structure, it is useful to have both upper and lower extrapolation results available. 6 T. HYTÖNEN AND C. PÉREZ 1.C. The A1theory. It is an interesting fact that if we assume that the weight satisfy the stronger condition w∈A1, then the estimate (1.2) can be considerably improved. Indeed, if Tis any Calderón–Zygmund operator, then Tis of course bounded on Lp(w), because A1⊂Apbut with a much better bound, namely kTkB(Lp(w)) ≤c pp′[w]A1,1< p < ∞.(1.19) Observe that the dependence on the A1constant is linear for any pwhile in the Ap case it is highly nonlinear for 1< p < 2, see (1.2). The result is sharp both in terms of the dependence on [w]A1, and in terms of the dependence on pwhen taking w= 1 by the classical theory. This fact was used to get the following endpoint result: kT fkL1,∞(w)≤c[w]A1log(e+ [w]A1)kfkL1(w).(1.20) See [27] and also [26] for these results and for more information about the problem. It was conjectured in [27] that the growth of this bound would be linear; however, it has been recently shown in [33] that the growth of the bound is worse than linear. It seems that most probably the Llog Lresult (1.20) is the best possible. On the other hand, in [28] a sort of “dual” estimate to the last bound was found, which is also of interest for related matters:   T f w  L1,∞(w)≤c[w]A1log(e+ [w]A1)ˆRd|f|dx. In this paper we improve these results following our new quantitative estimates involving this time A1and A∞control. To be precise, we will prove the following new results: 1.21. Theorem. Let Tbe a Calderón-Zygmund operator and let 1< p < ∞. Then kTkB(Lp(w)) ≤c pp′[w]1/p A1([w]′ A∞)1/p′ where c=c(d, T ). We will prove this by following the approach from [26, 27] to (1.19), modifying the proof at several points. In analogy to (1.5), Theorem 1.21 disproves the “reverse A1conjecture” [w]A1≤cTkTkB(Lp(w)) for all p∈(1,∞): considering a family of weights w∈A1for which [w]A∞grow slower than [w]A1, for any fixed Calderón– Zygmund operator T, we have (see Section 8 for details) inf w∈A1 kTkB(Lp(w)) [w]A1 = 0,1< p < ∞. Finally we will also use the approach from [27] and [28] to prove the following theorems respectively. 1.22. Theorem. Let Tbe a Calderón-Zygmund operator. Then kT fkL1,∞(w)≤cd,T [w]A1log(e+ [w]′ A∞)kfkL1(w). 1.23. Theorem. Let Tbe a Calderón-Zygmund operator. Then   T f w  L1,∞(w)≤cd,T [w]′ A∞log(e+ [w]A1)) kfkL1(Rd). SHARP WEIGHTED BOUNDS INVOLVING A∞7 1.D. Commutators with BMO functions. We further pursue the A∞point-of-view by proving a result in the spirit of Theorem 1.3 for commutators of linear operators Twith BMO functions. These operators are defined formally by the expression [b, T ]f=bT (f)−T(b f). More generally we can consider the kth order commutator defined by Tk b:= [b, T k−1 b]. When Tis a singular integral operator, these operators were considered by Coifman, Rochberg and Weiss [6] and since then many results have been obtaind. We refer to [4] for more information about these operators. It is shown in [4] that if T is a linear operator bounded on L2(w)for any w∈A2with bound kTkB(L2(w)) ≤ϕ([w]A2), where ϕis an increasing function ϕ: [1,∞)→[0,∞), then there is a dimensional constant csuch that k[b, T ]kB(L2(w)) ≤c ϕ(c[w]A2) [w]A2kbkBMO. In particular, if Tis any Calderón-Zygmund operator we can use the linear A2 theorem for Tto deduce k[b, T ]kB(L2(w)) ≤c[w]2 A2kbkBMO, and the quadratic exponent cannot be improved. An analogous result adapted to the A∞control reads as follows: 1.24. Theorem. Let Tbe a linear operator bounded on L2(w)for any w∈A2 and let b∈BMO. Suppose further that there is a function ϕ: [1,∞)3→[0,∞), increasing with respect to each component, such that kTkB(L2(w)) ≤ϕ[w]A2,[w]′ A∞,[σ]′ A∞. then there is a dimensional constant csuch that k[b, T ]kB(L2(w)) ≤c ϕc[w]A2, c [w]′ A∞, c [σ]′ A∞[w]′ A∞+ [σ]′ A∞kbkBMO, or more generally kTk bkB(L2(w)) ≤c ϕc[w]A2, c [w]′ A∞, c [σ]′ A∞[w]′ A∞+ [σ]′ A∞kkbkk BMO. We can now apply Theorem 1.3. 1.25. Corollary. Let Tbe any Calderón-Zygmund operator and let b∈BMO. Then k[b, T ]kB(L2(w)) ≤c[w]1/2 A2[w]′ A∞+ [w−1]′ A∞3/2kbkBMO, or more generally, kTk bkB(L2(w)) ≤c[w]1/2 A2[w]′ A∞+ [σ]′ A∞k+1/2kbkk BMO. 8 T. HYTÖNEN AND C. PÉREZ 1.E. An end-point estimate when p=∞. Having investigated the sharp bounds for Calderón-Zygmund operators T: Lp(w)→Lp(w)for p∈(1,∞)and w∈Ap, we finally consider the end-point T:L∞(w)→BMO(w), w ∈A∞. Qualitatively, this situation seems slightly uninteresting, as these end-point spaces simply reduce to their unweighted analogues: That L∞(w) = L∞with equal norms is immediate from the fact that wand the Lebesgue measure share the same zero sets for w∈A∞. That the weighted norm kfkBMO(w):= sup Q inf c 1 w(Q)ˆQ|f−c|w < ∞, is equivalent to the usual kfkBMO for w∈A∞was proven by Muckenhoupt and Wheeden [32], Theorem 5. However, one may still investigate the quantitative bound of operators T:L∞→BMO = BMO(w), when the latter space is equipped with the norm k kBMO(w). We start with: 1.26. Theorem. For w∈A∞, we have a bounded embedding I: BMO ֒→BMO(w) of norm at most c[w]′ A∞, where cis dimensional. This estimate is sharp in the following sense: if the norm of the embedding is bounded by φ([w]′ A∞), or just by φ([w]A∞), for all w∈A∞, then φ(t)≥ct. The following corollary for Calderón–Zygmund operators can be seen as an easy endpoint estimate of the bound kTkB(Lp(w)) ≤cp,T [w]Apfor p∈[2,∞). 1.27. Corollary. Let Tbe any Calderón-Zygmund operator and let w∈A∞, then T:L∞→BMO(w)with norm at most cT[w]′ A∞. Furthermore, this estimate is sharp in terms of the dependence on [w]′ A∞in the same way as Theorem 1.26. We conclude the introduction by stating the following observation which may be of some interest. 1.28. Proposition. If w∈A∞, then log w∈BMO with klog wkBMO ≤log(2e[w]A∞). 2. The two different A∞constants Before pursuing further our analysis of inequalities with A∞control, we include this short section to compare the two A∞constants [w]A∞:= sup Q− ˆQ wexp − ˆQ log w−1,[w]′ A∞:= sup Q 1 w(Q)ˆQ M(wχQ). We need the following auxiliary estimate, which is also used later in the paper: 2.1. Lemma. The logarithmic maximal function M0f:= sup Q exp − ˆQ log |f|χQ satisfies kM0fkLp≤c1/p dkfkLp for all p∈(0,∞). For the dyadic version, we can take cd=e, independent of dimension d. SHARP WEIGHTED BOUNDS INVOLVING A∞9 Proof. By Jensen’s inequality and the basic properties of the logarithm, we have M0f≤Mf, M0f= (M0|f|1/q)q≤(M|f|1/q)q, q ∈(0,∞), where Mis the Hardy–Littlewood maximal operator, or the dyadic maximal operator in the case of dyadic M0. By the Lqboundedness of the usual maximal function for q > 1, we have ˆ[M0f]p≤ˆ[M|f|p/q]q≤(Cd·q′)qˆ(|f|p/q)q= (Cd·q′)qˆ|f|p. In the non-dyadic case, we simply take, say, q= 2, giving the claim with cd= (2Cd)2. In the dyadic case, we have Cd= 1, and we can take the limit q→ ∞, which gives (q′)q=q q−1q=1 + 1 q−1q→e, and hence kM0fkp Lp≤ekfkp Lp. 2.2. Proposition. We have [w]′ A∞≤cd[w]A∞, where cdis as in Lemma 2.1. Proof. For x∈Q, it is not difficult to see that for the computation of M(wχQ)(x), it suffices to take the supremum over cubes R∋xwith R⊆Q: M(wχQ)(x) = sup R∋x R⊆Q− ˆR w, ∀x∈Q. By definition of [w]A∞, we have − ˆR w≤[w]A∞exp − ˆR log w, and hence, taking the supremum over R, M(wχQ)(x)≤[w]A∞M0(wχQ)(x),∀x∈Q. Integration over Qand application of Lemma 2.1 now give ˆQ M(wχQ)≤[w]A∞ˆM0(wχQ)≤[w]A∞cdˆwχQ=cd[w]A∞w(Q); thus [w]′ A∞≤cd[w]A∞. It is a well known fact that any A∞weight satisfies a reverse reverse Hölder inequality playing a central role in the area. In this paper a sharp version of this property will also play a fundamental role. To be precise if w∈A∞we define r(w) := 1 + 1 τd[w]′ A∞ where τdis a dimensional constant that we may take to be τd= 211+d. Note that r(w)′≈[w]′ A∞. The result we need is the following. 2.3. Theorem (A new sharp reverse Hölder inequality). a) If w∈A∞, then − ˆQ wr(w)1/r(w)≤2− ˆQ w. 16 T. HYTÖNEN AND C. PÉREZ Proof of Theorem 1.10. Consider the 2dshifted dyadic systems Dα:= {2−k[0,1)d+m+ (−1)kα:k∈Z, m ∈Zd}, α ∈ {0,1 3}d. One can check (perhaps best in dimension n= 1 first) that any cube Qis contained in a shifted dyadic cube Qα∈Dαwith ℓ(Qα)≤6ℓ(Q), for some α. Hence − ˆQ|f| ≤ 6d− ˆQα|f| ≤ 6dMα df, and therefore Mf ≤6dX α∈{0,1 3}d Mα df. Thus the norm bound for Mdmay be multiplied by 12dto give a bound for M. 4.4. Remark. A recent result of the first author and A. Kairema [17] allows to perform a similar trick with adjacent dyadic systems even in an abstract space of homogeneous type. Thus, Corollary 1.10 readily extends to this generality as well. 4.B. Proof of Theorem 4.3. We start by observing that it suffices to have a uniform bound over all linearizations ˜ M(fσ) = X Q∈D χE(Q)hfσiQ, where the sets E(Q)⊆Qare pairwise disjoint. Here we using the following notation hfiQ=− ˆQ f=− ˆQ f(x) dx and hfiσ Q=1 σ(Q)ˆQ f(x)σ(x) dx where as usual σ(E) = ´Qσ(x) dx By this disjointness, k˜ M(fσ)kLp(w)=X Q∈D w(E(Q))hfσip Q1/p =X Q∈D w(E(Q))σ(Q) |Q|p(hfiσ Q)p1/p Now recall: 4.5. Theorem (Dyadic Carleson embedding theorem).Suppose that the nonnegative numbers aQsatisfy X Q⊆R aQ≤Aσ(R)∀R∈D. Then, for all p∈[1,∞)and f∈Lp(σ), X Q∈D aQ(hfiσ Q)p1/p ≤A1/pkMd,σfkLp(σ) ≤A1/p ·p′·kfkLp(σ)if p > 1. SHARP WEIGHTED BOUNDS INVOLVING A∞17 Since this is a slightly nonstandard formulation, although immediate by inspection of the usual argument, we provide a proof for completeness: Proof. We view the sum PQaQ(hfiQ)pas an integral on a measure space (D, µ) built over the set of dyadic cubes D, assigning to each Q∈Dthe measure aQ. Thus X Q∈D aQ(hfiQ)p=ˆ∞ 0 pλp−1µ({Q∈D:hfiQ> λ}) dλ =: ˆ∞ 0 pλp−1µ(Qλ) dλ Let Q∗ λbe the set of maximal dyadic cubes Rwith the property that hfiR> λ. The cubes R∈Q∗ λare disjoint, and their union is equal to the set {Md,σf > λ}. Thus µ(Qλ) = X Q∈Qλ aQ≤X R∈Q∗ λX Q⊆R aQ≤X R∈Q∗ λ Aσ(R) = Aσ(Md,σf > λ), and hence X Q∈D aQ(hfiQ)p≤Aˆ∞ 0 pλp−1σ(Md,σf > λ) dλ=AkMd,σfkp Lp(σ). If we apply the Carleson embedding with aQ=w(E(Q))σ(Q)/|Q|p, we find that k˜ M(fσ)kLp(w)≤A1/pkMd,σfkLp(σ)(4.6) provided that X Q⊆R w(E(Q))σ(Q) |Q|p≤A σ(R)∀R∈D.(4.7) Note that on E(Q)⊆Q⊆R, we have σ(Q)/|Q| ≤ M(σχR), and hence X Q⊆R w(E(Q))σ(Q) |Q|p=ˆX Q⊆R χE(Q)σ(Q) |Q|pw ≤ˆX Q⊆R χE(Q)M(χRσ)pw≤ˆR M(χRσ)pw. So if kχRM(χRσ)kLp(u)≤A1/p σ(R)1/p, then (4.7) holds, hence by Carleson’s embedding also (4.6), and therefore the original two-weight inequality kM(fσ)kLp(u)≤A1/p kMd,σfkLp(σ). Hence, we are reduced to proving that kχRM(χRσ)kp Lp(u)≤A σ(R), A = (4e)1/p ·Bp[w, σ].(4.8) (In fact, the argument up to this point was essentially reproving Sawyer’s two-weight characterization for the maximal function, paying attention to the constants.) To prove (4.8), we exploit another linearization of Minvolving the principal cubes, as in the proof of the A2theorem: Let S0:= {R}and recursively Sk:= [ S∈Sk−1{Q⊂S:hσiQ>2hσiS, Q is a maximal such cube}, 18 T. HYTÖNEN AND C. PÉREZ and then S:= S∞ k=0 Sk. The pairwise disjoint subsets E(S)⊆S, defined in (3.12), satisfy |E(S)| ≥ 1 2|S|by (3.13), and they partition R. If x∈E(S)and Q∋x, then hσiQ≤2hσiS, and hence χRM(χRσ)≤2hσiSon χE(S). So altogether kχRM(χRσ)kp Lp(w)≤2p  X S∈S χE(S)hσiS   p Lp(w) = 2pX S∈S w(E(S))σ(S) |S|p ≤2pX S∈S w(S) |S|σ(S) |S|p|S| ≤2p+1 X S∈S Bp[w, σ] exp − ˆS log σ|E(S)| ≤2p+1Bp[w, σ]ˆRX S∈S exp − ˆS log σχE(S) ≤2p+1Bp[w, σ]ˆR sup Q∈D χQexp − ˆQ log σχR = 2p+1Bp[w, σ]ˆR M0(χRσ), (4.9) where M0is the (dyadic) logarithmic maximal function introduced in Lemma 2.1. By this lemma, we then have kχRM(χRσ)kp Lp(u)≤4pBp[w, σ]·e·σ(R), which proves (4.8), and hence Theorem 4.3, upon taking the pth root. In order to prove the second version of Theorem 4.3, we only need to make a slight modification in the estimate (4.9). We then compute: kχRM(χRσ)kp Lp(w)≤2pX S∈S w(S) |S|σ(S) |S|p|S| ≤2p+1 X S∈S [w]Ap σ(S) |S||E(S)| ≤2p+1[w]ApX S∈SˆE(S) M(σχQ) = 2p+1[w]ApˆQ M(σχQ) = 2p+1[w]Ap[σ]′ A∞σ(Q), by a direct application of the definition of [σ]′ A∞in the last step, and this completes the alternative argument. 4.C. Another proof of Theorem 4.3. We finish this section by providing yet another proof variant for Theorem 4.3. This proof is more elementary, since it does not need the reduction to the testing condition (4.8), and it uses the more standard Calderón–Zygmund-type stopping SHARP WEIGHTED BOUNDS INVOLVING A∞19 cubes, instead of the principal cubes. Its disadvantage is the fact the we cannot recover the dimension-independence by this argument. On the other hand, the proof may be extended to maximal functions defined in term of a general basis (see [13] Section IV.4). A simpler proof of Theorem 4.3 with a dimension-dependent bound. Fix a > 2d. For each integer klet Ωk={x∈Rd:Md(f σ)(x)> ak}. By standard arguments we consider the Calderón–Zygmund decomposition and there is a family of maximal non-overlapping dyadic cubes {Qk,j}for which Ωk= SjQk,j and ak<1 |Qk,j|ˆQk,j |f(y)|σ(y)dy ≤2dak.(4.10) Now, ˆRd Md(fσ)pwdx=X kˆΩk\Ωk+1 Md(fσ )pwdx ≤apX k akpw(Ωk) = apX k,j akpw(Qk,j ) ≤apX k,j 1 |Qk,j|ˆQk,j |f(y)|σ(y) dy!p w(Qk,j ) =apX k,j h|f|iσ Qk,j pσ(Qk,j ) |Qk,j|p w(Qk,j ) ≤apBp[w, σ]X k,j h|f|iσ Qk,j p|Qk,j |exp − ˆQk,j log σ(t) dt =apBp[w, σ]X Q∈Dh|f|iσ QpaQ, where aQ=(|Q|exp − ´Qlog σif Q=Qk,j for some (k, j), 0else. By the dyadic Carleson embedding theorem, we can hence conclude that ˆRd Md(fσ)pwdx≤apBp[w, σ]AˆRd (Md,σf)pσdx, provided that we check the condition X Q⊆R aQ=X k,j:Qk,j ⊆R|Qk,j |exp − ˆQk,j log σ≤A|R|.(4.11) To estimate the left side of (4.11), we do first the following: For each (k, j)we set Ek,j =Qk,j \Ωk+1. Observe that the sets of the family Ek,j are pairwise disjoint. We claim that |Qk,j|<a a−2d|Ek,j |(4.12) 20 T. HYTÖNEN AND C. PÉREZ for each k, j. Indeed, by (4.10) and Hölder’s inequality, |Qk,j ∩Ωk+1|=X Qk+1,l⊂Qk,j |Qk+1,l| <1 ak+1 X Qk+1,l⊂Qk,j ˆQk+1,l |f|σ ≤1 ak+1 ˆQk,j |f|σ≤2d a|Qk,j|, which proves (4.12). With β=a a−2dwe can estimate the left side of (4.11) as follows: X Q⊆R aQ≤βX (k,j):Qk,j ⊆R|Ek,j|exp − ˆQk,j log σ(t)dt ≤βX (k,j):Qk,j ⊆RˆEk,j M0(σ1R)(x) dx ≤βˆR M0(σ1R)(x) dx≤β e σ(R), where we used the definition and the L1boundedness of the logarithmic dyadic maximal function. This proves (4.11) with A=β e, concluding the proof.  5. Proof of the extrapolation theorems We will prove in this section the Upper and Lower Extrapolation Theorems 1.17 and 1.18. Recall that the initial hypothesis is given by the expression: kT fkLr(w)≤ϕ[w]Ar,[w]A∞,[w−1/(r−1)](r−1) A∞kfkLr(w) for some r∈(1,∞). Proof of Theorem 1.17. Our argument is modeled after a simplified proof of the Dragičević–Grafakos–Pereyra–Petermichl [10] result due to Duoandikoetxea [12] (see also [9]). Fix some p∈(1, r),w∈Ap,f∈Lp(w)and g:= |f|/kfkLp(w). Let Rg := ∞ X k=0 2−kMkg kMkk B(Lp(w)) so that |g| ≤ Rg, kRgkLp(w)≤2kgkLp(w)= 2,[Rg]A1≤2kMkLp(w). Then by Hölder’s inequality kT fkLp(w)=ˆ|T f|p(Rg)−(r−p)p/r(Rg)(r−p)p/rw1/p ≤ˆ|T f|r(Rg)−(r−p)w1/rˆ(Rg)pw1/p−1/r ≤ kT fkLr(W)(2p)1/p−1/r ≤2kT fkLr(W), W := (Rg)−(r−p)w. SHARP WEIGHTED BOUNDS INVOLVING A∞21 By assumption, we have kT fkLr(W)≤ϕ[W]Ar,[W]A∞,[W−1/(r−1)](r−1) A∞kfkLr(W) where kfkLr(W)=ˆ|f|r(Rf)−(r−p)w1/rkfk(r−p)/r Lp(w) ≤ˆ|f|r|f|−(r−p)w1/rkfk(r−p)/r Lp(w)=kfkLp(w), so it remains to estimate the weight constants [W]Ar,[W]A∞,[W−1/(r−1)]A∞. Using supQ(Rg)−1≤[Rg]A1hRgi−1 Qor Hölder’s or Jensen’s inequality where appropriate, we compute hWiQ=h(Rg)−(r−p)wiQ ≤[Rg]r−p A1hRgi−(r−p) QhwiQ, hW−1/(r−1)ir−1 Q=h(Rg)(r−p)/(r−1)w−1/(r−1)ir−1 Q ≤ hRgir−p Qhw−1/(p−1)ip−1 Q, exph−log WiQ=exphlog(Rg)iQr−pexph−log wiQ ≤ hRgir−p Qexph−log wiQ, and exph−log W−1/(r−1)iQr−1 =exphlog(Rg)−1iQr−pexph−log w−1/(r−1)iQr−1 ≤[Rg]r−p A1hRgi−(r−p)exph−log w−1/(p−1)iQp−1. Multiplying the appropriate estimates and using the definition, we then have [W]Ar≤[Rg]r−p A1[w]Ap,[W]A∞≤[Rg]r−p A1[w]A∞, [W−1/(r−1)]r−1 A∞≤[Rg]r−p A1[w−1/(p−1)]p−1 A∞. (We do not know whether it is possible to make similar estimates for [W]′ A∞in terms of [w]′ A∞; this is the reason why we need to use the [ ]A∞constants in this proof.) Next, recall that [Rg]A1≤2kMkB(Lp(w)) ≤cd·p′·[w]1/p Ap([w−1/(p−1)]′ A∞)1/p. Thus we conclude the proof with kT fkLp(w)≤2kT fkLr(W)≤2ϕ[W]Ar,[W]A∞,[W−1/(r−1)](r−1) A∞kfkLr(W) ≤2ϕ[Rg]r−p A1[w]Ap,[w]A∞,[w−1/(p−1)](p−1) A∞kfkLp(w) ≤2ϕ2r−pkMkr−p B(Lp(w))[w]Ap,[w]A∞,[w−1/(p−1)](p−1) A∞kfkLp(w). 22 T. HYTÖNEN AND C. PÉREZ Proof of Theorem 1.18. Again, our argument is inspired by a simplified proof of the Dragičević–Grafakos–Pereyra–Petermichl [10] result due to Duoandikoetxea [12] (see also [9]). Fix some p∈(r, ∞),w∈Ap,f∈Lp(w). By duality, we need have kT fkLp(w)= sup h≥0 khkLp′(w)=1 ˆ|T f|hw. We fix one such h, and try to bound the expression on the right. Observe that the pointwise multiplication operators h7→ wh :Lp′(w)→Lp′(w1−p′), g 7→ 1 wg:Lp′(w1−p′)→Lp′(w) are isometric. Let Rbe as in the previous proof, except with p′and σ=w1−p′in place of pand w: Rg := ∞ X k=0 2−kMkg kMkk B(Lp′(σ)) , and R′h:= w−1R(wh). Then h≤R′h, kR′hkLp′(w)≤2khkLp′(w)= 2,[wR′h]A1≤2kMkB(Lp′(σ)). Then by Hölder’s inequality ˆ|T f|hw ≤ˆ|T f|(R′h)w=ˆ|T f|(R′h)(p−r)/[r(p−1)](R′h)(r−1)p/[r(p−1)]w ≤ˆ|T f|r(R′h)(p−r)/(p−1)w1/rˆ(R′h)p/(p−1)w1/r′ ≤ kT fkLr(W)2p′/r′, W := (R′h)(p−r)/(p−1)w. By assumption, kT fkLr(W)≤ϕ[W]Ar,[W]A∞,[W−1/(r−1)](r−1) A∞kfkLr(W)(5.1) where, by Hölder’s inequality with exponents p/r and its p/(p−r), kfkLr(W)=ˆ|f|rwr/p ·(R′h)(p−r)/(p−1)w(p−r)/p1/r ≤ˆ|f|pw1/pˆ(R′h)p/(p−1)w1/r−1/p ≤ kfkLp(w)(2p′)1/r−1/p, so altogether, supressing the arguments of ϕfrom (5.1), ˆ|T f|hw ≤ kT fkLr(W)2p′/r′≤ϕ...kfkLr(W)2p′/r′ ≤ϕ...(2p′)1/r−1/pkfkLp(w)2p′/r′= 2ϕ(...)kfkLp(w). It remains to estimate [W]Ar,[W]A∞,[W−1/(r−1)](r−1) A∞ for W= (R′h)(p−r)/(p−1)w= [(R′h)w](p−r)/(p−1)w(r−1)/(p−1). SHARP WEIGHTED BOUNDS INVOLVING A∞23 We thus compute hWiQ=h(R′h)(p−r)/(p−1)wiQ ≤ h(R′h)wi(p−r)/(p−1) Qhwi(r−1)/(p−1) Q, hW−1/(r−1)ir−1 Q=h(wR′h)−(p−r)/[(p−1)(r−1)]w−1/(p−1)ir−1 Q ≤[wR′h](p−r)/(p−1) A1h(R′h)wi−(p−r)/(p−1) Qhw−1/(p−1)ir−1 Q, exp(−hlog WiQ) = exphlog(wR′h)−1iQ(p−r)/(r−1)exph−log wiQ(r−1)/(p−1) ≤[(R′h)w](p−r)/(r−1) A1h(R′h)wi−(p−r)/(r−1) Q ×exph−log wiQ(r−1)/(p−1), and exp(−hlog W−1/(r−1)iQ)r−1 =exp(hlog(wR′h)iQ)(p−r)/(p−1)exph−log w−1/(p−1)iQr−1 ≤ h(R′h)wi(p−r)/(r−1) Qexph−log w−1/(p−1)iQr−1. Multiplying the relevant quantities, it follows that [W]Ar≤[(R′h)w](p−r)/(p−1) A1[w](r−1)/(p−1) Ap, [W]A∞≤[(R′h)w](p−r)/(p−1) A1[w](r−1)/(p−1) A∞, [W−1/(r−1)]r−1 A∞≤[(R′h)w](p−r)/(p−1) A1[w−1/(p−1)](r−1) A∞, Also recall that [(R′h)w]A1≤2kMkB(Lp′(w1−p′)) ≤cd[w1−p′]1/p′ Ap′[w]1/p′ A∞=cd[w]1/p Ap[w]1/p′ A∞, and thus we conclude with kT fkLp(w)≤ˆ|T f|hw ≤2ϕ[W]Ar,[W]A∞,[W−1/(r−1)](r−1) A∞kfkLp(w) ≤2ϕ[(R′h)w](p−r)/(p−1) A1[W]Ar,[W]A∞,[W−1/(r−1)](r−1) A∞kfkLp(w) ≤2ϕ(2kMkB(Lp′(w1−p′)))(p−r)/(p−1) ×[w](r−1)/(p−1) Ar,[w](r−1)/(p−1) A∞,[w−1/(p−1)](r−1) A∞kfkLp(w). 6. The A1theory, proof of Theorem 1.21 and its consequences 6.A. The main lemma. The proofs of the theorems will be based on the following lemma. 6.1. Lemma. Let Tbe any Calderón-Zygmund singular integral operator and let w be any weight. Also let p, r ∈(1,∞). Then, there is a constant c=cd,T such that: kT fkLp(w)≤cpp′(r′)1/p′kfkLp(Mrw) where as usual we denote Mrw=M(wr)1/r. 24 T. HYTÖNEN AND C. PÉREZ This is a consequence of the following estimate that can be found in [27] when r∈(1,2]: kT fkLp(w)≤cpp′1 r−11−1/pr kfkLp(Mrw) since 1 r−11−1/pr ≤(r′)1−1/p+1/pr′≤2(r′)1/p′ and t1/t ≤2,t≥1. 6.B. Proof of the sharp reverse Hölder’s inequality. We need the following lemma: 6.2. Lemma. For any cube Qand any measurable function w, ˆQ wlog(e+w hwiQ ) dx≤2d+1 ˆQ M(wχQ) dx, (6.3) Hence, if w∈A∞ sup Q 1 w(Q)ˆQ w(y) log(e+w(y) hwiQ )dy ≤2d+1 [w]′ A∞(6.4) The essential idea of the proof can be traced back to the well known Llog L estimate for Min [45]. However these estimates are not homogeneus. A proof of this lemma within the context of spaces of homogeneous type can essentially be found in [38, Lemma 8.5] (see also [50, p. 17, inequality (2.15)] for a different proof). Proof of Lemma 6.2. Fix a cube Q. By homogeneity we assume that hwiQ= 1. The key estimate follows from the “reverse weak type (1,1) estimate”: if wis nonnegative and t > hwiQ, 1 tˆ{x∈Q:w(x)>t} wdx≤2d|{x∈Q:M(wχQ)(x)> t}|.(6.5) Now, 1 |Q|ˆQ wlog(e+w) dx=1 |Q|ˆ∞ 0 1 e+tw({x∈Q:w(x)> t}) dt =1 |Q|ˆ1 0 +1 |Q|ˆ∞ 1···=I+II, and I≤1≤1 |Q|ˆQ M(wχQ) dx. For II we use estimate (6.5): II =1 |Q|ˆ∞ 1 1 e+tw({x∈Q:w(x)> t}) dt ≤2d |Q|ˆ∞ 1 t e+t|{x∈Q:M(wχQ)(x)> t}|dt ≤2d |Q|ˆ∞ 0|{x∈Q:M(wχQ)(x)> t}|dt =2d |Q|ˆQ M(wχQ)(x) dx. SHARP WEIGHTED BOUNDS INVOLVING A∞25 This gives (6.3) and (6.4) follows from the definiton of [w]′ A∞. The main use of the Lemma is the following key observation that we borrow from [50], p. 45: 6.6. Lemma. Let S⊂Qand let λ > 0, then |S| |Q|< e−λimplies w(S) w(Q)<2d+2[w]′ A∞ λ+e−λ/2(6.7) Proof. Indeed, if Eλ={x∈Q:w(x)> eλhwiQ}then w(Eλ)≤2d+1 λw(Q)by (6.4). Therefore: w(S)≤w(S∩Eλ/2) + w(S\Eλ/2)≤2d+2 [w]′ A∞ λw(Q) + eλ/2hwiQ|S| ≤2d+2 [w]′ A∞ λw(Q) + eλ/2e−λw(Q)by the hypothesis in (6.7) =2d+2 [w]′ A∞ λw(Q) + e−λ/2w(Q) and this proves the claim (6.7).  Proof of Theorem 2.3. Recall that we have to prove that − ˆQ wr(w)1/r(w)≤2− ˆQ w. where r(w) := 1 + 1 τd[w]′ A∞ , and where τdis a large dimensional constant. Observe that by homogeneity we can assume that − ´Qw= 1. We use the dyadic maximal function on the dyadic subcubes of a given Q: ˆQ w1+ε≤ˆQ Md(wχQ)εw=ˆ∞ 0 εtε−1w({x∈Q:Md(wχQ)> t}) dt. ≤ˆ1 0 εtε−1w(Q) dt+εˆ∞ 1 εtεw({x∈Q:Md(wχQ)> t})dt t ≤ |Q|+εX k≥0ˆak+1 ak tεw({x∈Q:Md(wχQ)> t})dt t ≤ |Q|+εaεX k≥0 akε ˆak+1 ak w({x∈Q:Md(wχQ)> ak})dt t,for a≫1, =|Q|+εaεlog aX k≥0 akε w(Ωk) where Ωk={x∈Q:Md(wχQ(x)> ak}. Since ak≥1 = − ´Qwwe can consider the Calderón–Zygmund decomposition w adapted to Q. There is a family of maximal non-overlapping dyadic cubes {Qk,j } 32 T. HYTÖNEN AND C. PÉREZ 7.4. Lemma. There are dimensional constants ǫdand cdsuch that [eRe zbw]′ A∞≤cd[w]′ A∞if |z| ≤ ǫd kbkBMO[w]′ A∞ . Proof. We know that wsatisfies the reverse Hölder inequality − ´Qw1+3δ1/(1+3δ)≤ 2− ´Qwwith a constant δ=cd/[w]′ A∞<2−1, where cdis a small dimensional constant. We will prove that eRe zbwsatisfies a reverse Hölder estimate − ˆQ (eRe zbw)1+δ1/(1+δ)≤Cd− ˆQ eRe zbw, (7.5) for all zas in the assertion. This shows that [eRe zbw]′ A∞≤2Cd/δ ≤cd[w]′ A∞ by part b) of Theorem 2.3. To prove (7.5), we first have − ˆQ (eRe zbw)1+δ1/(1+δ)=eRe zhbiQ− ˆQ (eRe z(b−hbiQ)w)1+δ1/(1+δ) ≤eRe zhbiQ− ˆQ eRe z(b−hbiQ)(1+δ)2/δδ/(1+δ)2− ˆQ w(1+δ)21/(1+δ)2 , where we applied Hölder’s inequality with exponents (1 + δ)/δ and 1 + δ. Now (1 + δ)2= 1 + 2δ+δ2≤1 + 3δ, and hence the last factor is bounded by 2− ´Qw. Moreover, by Lemma 7.1, we have − ˆQ eRe z(b−hbiQ)(1+δ)2/δ ≤βdif |z| ≤ αdδ 4kbkBMO So altogether − ˆQ (eRe zbw)1+δ1/(1+δ)≤eRe zhbiQ·βd·2− ˆQ w, (7.6) and we concentrate on the last factor. We observe that − ˆQ w2=− ˆQ w(1+δ)/2w(1−δ)/22 ≤− ˆQ w1+δ− ˆQ w1−δ≤2− ˆQ w1+δ− ˆQ w1−δ, and hence − ˆQ w≤2(1+δ)/(1−δ)− ˆQ w1−δ1/(1−δ) ≤8− ˆQ w1−δeRe zb(1−δ)e−Re zb(1−δ)1/(1−δ) ≤8− ˆQ weRe zb− ˆQ e−Re zb(1−δ)/δδ/(1−δ), where we used Hölder’s inequality with exponents 1/(1 −δ)and 1/δ. SHARP WEIGHTED BOUNDS INVOLVING A∞33 Combining with (7.6), we have shown that − ˆQ (eRe zbw)1+δ1/(1+δ) ≤eRe zhbiQ·βd·16− ˆQ weRe zb− ˆQ e−Re zb(1−δ)/δδ/(1−δ) = 16βd·− ˆQ weRe zb− ˆQ e−Re z(b−hbiQ)(1−δ)/δδ/(1−δ) ≤16βd·− ˆQ weRe zb·βd, provided that |z| ≤ αdδ/kbkBMO in the last step. Altogether, we have proven (7.5) with Cd= 16β2 d, under the condition that |z| ≤ αdδ/(4kbkBMO), and this completes the proof.  Proof of Theorem 1.24. The proof is a revised version of that of [4] following the second proof in the classical Lptheorem for commutators that can be found in [6]. Indeed, we begin by considering the “conjugate” of the operator given by Tz(f) = ezbT(e−zbf). where zis any complex number. Then, a computation gives (for instance for “nice” functions), [b, T ](f) = d dz Tz(f)|z=0 =1 2πi ˆ|z|=ǫ Tz(f) z2dz , ǫ > 0, by the Cauchy integral theorem. Now, by Minkowski’s inequality k[b, T ](f)kL2(w)≤1 2π ǫ2ˆ|z|=ǫkTz(f)kL2(w)|dz|, ǫ > 0,(7.7) all we need to do is estimate kTz(f)kL2(w)=kT(e−zbf)kL2(e2 Re z bw),for |z|=ǫ with appropriate ǫ. By the main hypothesis of the theorem, we have kT(e−zbf)kL2(w)≤ϕ[e2 Re z bw]A2,[e2 Re z bw]′ A∞,[e2 Re z bσ]′ A∞ke−zbfkL2(e2 Re z bw), where ke−zbfkL2(e2 Re z bw)=kfkL2(w). By Lemmas 7.3 and 7.4 (the latter applied to both wand w−1), we have [we2 Re bz]A2≤Cd[w]A2,[we2 Re bz]′ A∞≤Cd[w]′ A∞, [w−1e−2 Re bz]′ A∞≤Cd[w−1]′ A∞, provided that |z|=ǫ≤ǫd kbkBMO[w]′ A∞+ [w−1]′ A∞ Using this radius and the above estimates in (7.7), we obtain k[b, T ](f)kL2(w)≤1 2πǫ2ˆ|z|=ǫ ϕCd[w]A2, Cd[w]′ A∞, Cd[w−1]′ A∞kfkL2(w)|dz| ≤CdkbkBMO[w]′ A∞+ [w−1]′ A∞ ×ϕCd[w]A2, Cd[w]′ A∞, Cd[w−1]′ A∞kfkL2(w). This concludes the proof of the main part of the theorem. The estimate for Tk b is deduced by iterating from the case k= 1. 34 T. HYTÖNEN AND C. PÉREZ 8. Examples We compare our new estimates with earlier quantitative results by means of some examples. 8.A. Power weights and the maximal inequality. Let d= 1 and p∈(1,∞)be fixed; we do not pay attention to the dependence of multiplicative constants on p. For w(x) = |x|αand −1< α < p −1, one easily checks that [w]Aph1 1 + α·1 ((p−1) −α)p−1, [w]A∞h1 1 + α,[w−1/(p−1)]A∞h1 (p−1) −α; moreover, the functionals [ ]A∞and [ ]′ A∞are comparable for these weights. Letting α→ −1or α→p−1, this shows that we have power weights with [w]Ap=t≫1and either [w]A∞htand [w−1/(p−1)]A∞h1, or [w]A∞h1and [w−1/(p−1)]A∞ht1/(p−1). With [w]Aph[w]A∞ht≫1and [w−1/(p−1)]A∞h1, our maximal estimate kMkB(Lp(w)) .[w]Ap[w−1/(p−1)]A∞1/p ht1/p clearly improves on Buckley’s bound kMkB(Lp(w)) .[w]1/(p−1) Ap ht1/(p−1). Despite this improvement over earlier estimates, our bounds fail to provide a twosided estimate for the norm of the maximal operator: A. Lerner and S. Ombrosi [25] have constructed a family of weights which shows that inf w∈A2 kMkB(L2(w)) [w]A2[w−1]′ A∞1/2= 0. The weights of their example are products of power weights and the two-valued weights considered in the next subsection. 8.B. Two-valued weights and Calderón–Zygmund operators. The estimates for the Muckenhoupt constants of power weights in the previous subsection show that [w]A2h[w]A∞+ [w−1]A∞h[w]′ A∞+ [w−1]′ A∞for w(x) = |x|α, d = 1, so the improvement of our bound kTkB(L2(w)) .[w]1/2 A2[w]′ A∞+ [σ]′ A∞1/2 over kTkB(L2(w)) .[w]A2is invisible to such weights. However, the difference can be observed with weights of the form w=t·χE+ χR\E, where t > 0and E⊂Ris a measurable set, so that both Eand R\Ehave positive Lebesgue measure. As Iranges over all intervals of R, the ratio |E∩I|/|I| ranges (at least) over all values α∈(0,1), and hence [w]A2= sup α∈(0,1) (αt + 1 −α)(αt−1+ 1 −α) = (t+ 1)2 4t, SHARP WEIGHTED BOUNDS INVOLVING A∞35 and [w]A∞= sup α∈(0,1) (αt + 1 −α)e−αlog t=: sup α∈(0,1) f(α). Now f′(α) = 0 at the unique point ˆα= 1/log t−1/(t−1) ∈(0,1), and so [w]A∞=f(ˆα) = e−1t−1 log texp log t t−1h(t/ log t, t ≫1, t−1/log t−1,0< t ≪1. Assume then that t≫1so that [w]A∞ht/ log t. Since σis a weight of the same form with t−1≪1in place of t, we also have [σ]A∞ht/ log t. Thus [w]A2ht, kTkB(L2(w)) .[w]1/2 A2[w]A∞+ [σ]A∞1/2ht √log t. In particular, the above estimates already show that inf w∈A2 kTkB(L2(w)) [w]A2 = 0. If we use the sharper version of our A2theorem with the weight constants [ ]′ A∞ instead, we find that kTkB(L2(w)) actually grow much slower than [w]A2: 8.1. Lemma. For w=t·χE+χR\Eand t≥3, we have [w]′ A∞≤4 log t. (With the earlier estimate for [w]A∞, this shows that [w]A∞can be exponentially larger than [w]′ A∞.) Proof. Note that χIM(wχI) = χIsup J⊆I χJ− ˆJ w=χIsup J⊆I χJ 1 |J|(|J\E|+t|J∩E|) =χIsup J⊆I χJ1 + (t−1)|J∩E| |J|=χI1 + (t−1)M(χI∩E), and hence, abbreviating τ:= t−1, ˆI M(wχI) = |I|+τˆI M(χI∩E) = |I|+τˆ1 0|I∩{M(χI∩E)> λ}|dλ ≤ |I|+τˆa 0|I|dλ+ˆ1 a 2 λ|I∩E|dλ =|I|+τa|I|+ 2|I∩E|log 1 a =|I|+τ|I∩E|1 + 2 log |I| |I∩E|, a := |I∩E| |I|, where the factor 2is the weak-type (1,1) norm of the maximal operator on the real line. Since w(I) = |I|+τ|I∩E|, we have [w]′ A∞= sup I 1 w(I)ˆI M(wχI)≤sup α∈(0,1) 1 + τα(1 + 2 log α−1) 1 + τα = 1 + 2 sup α∈(0,1) τα 1 + τα log 1 α, (8.2) 36 T. HYTÖNEN AND C. PÉREZ recalling that the ratio |I∩E|/|I|attains at least all values α∈(0,1) as Iranges over all intervals. If α≥τ−1, then log α−1≤log τ, while τα/(1 + τα)≤1. If α≤τ−1, then τα log 1 α=τα log 1 τα +τα log τ≤1 e+ log τ, as xlog x−1≤e−1and x≤1for x=τα ∈(0,1). Altogether, recalling that t=τ+ 1 ≥3, we have [w]′ A∞≤1 + 21 e+ log τ)≤(1 + 2 e) + 2 log t≤4 log t.  Since σ=w−1is a weight of the same form, we find that for these particular weights, [w]A2ht, kTkB(L2(w)) .[w]1/2 A2[w]′ A∞+ [σ]′ A∞1/2.tlog t)1/2, so indeed kTkB(L2(w)) can grow much slower than [w]A2for such particular families of weights. This example also motivates the use of the A∞constants [w]′ A∞, rather than [w]A∞, whenever this is possible. In a similar way we can show that the main result from Theorem 1.21 strictly improves on the earlier estimate (1.19). Indeed, if we let wbe the previous weight with t≫1so that wA1htand [w]A∞ht/ log t, then [w]1/p A1[w]1/p′ A∞ ht (log t)1/p′. As above, this family of weights shows that inf w∈A1 kTkB(Lp(w)) [w]A1 = 0,1< p < ∞. 8.C. Two-valued weights and dyadic shifts. Although it was not stated explicitly above, from the proof it is clear that our weighted bound for the dyadic shifts only depends on the dyadic Muckenhoupt constants, where the supremum is over dyadic cubes only, instead of all cubes. This makes a difference for the two-valued weights w=t·χE+χR\Econsidered above, when the set Eis appropriately chosen. Indeed, with E:= Sk∈Z[2k, 2k+1), one observes that the ratio |E∩I|/|I|only attains the values 0,1 2,1as Iranges over the dyadic intervals. Consequently, the dyadic A∞constant has a different expression [w]d A∞= max α∈{0,1 2,1} (αt + 1 −α)e−αt =t+ 1 2√t= ([w]d A2)1/2, where [w]d A2= [w]A2, as one easily observes. Repeating the proof of Lemma 8.1 in the dyadic case (recalling that the weak-type (1,1) norm is Cd= 1 for the dyadic maximal operator), we get in place of (8.2) that [w]′,d A∞≤1 + sup α∈{0,1/2,1} τα 1 + τα log 1 α= 1 + 1 2τ 1 + 1 2τlog 2 ≤1 + log 2. So these constants are actually uniformly bounded over the choice of the parameter t. SHARP WEIGHTED BOUNDS INVOLVING A∞37 By symmetry, we also have [w−1]d A∞= [w]d A∞and [w−1]′,d A∞= [w]′,d A∞, and hence, for this particular Eand w=t·χE+χR\E, kXkB(L2(w)) .(r+ 1)2[w]d A21/2[w]′,d A∞+ [w−1]′,d A∞1/2.(r+ 1)2[w]d A21/2. The first A∞constants [ ]d A∞would have given the weaker bound kXkB(L2(w)) . (r+ 1)2[w]d A23/4, instead. 8.D. The extrapolated bounds for Calderón–Zygmund operators. It is interesting to compare our estimate (1.8), namely kTkB(Lp(w)) .[w]2/p−1/[2(p−1)] Ap[w]1/[2(p−1)] A∞+ [σ]1/2 A∞([w]′ A∞)1−2/p .[w]2/p Ap([w]′ A∞)1−2/p,(8.3) which is valid for any Calderón–Zygmund operator and for all p≥2, with an estimate implicitly contained in the proof of a related result by Lerner [24], Theorem 1.2. He considers maximal trunctations T∗of convolution-type Calderón– Zygmund operators, and obtains the following bound: kT∗kB(Lp(w)) .[w]1/2 Ap([w]′ A∞)1/2+kMkB(Lp(w)), .[w]1/2 Ap([w]′ A∞)1/2, p ∈[3,∞),(8.4) where the second estimate is an application of Buckley’s result (we do not even need our improvement at this point), kMkB(Lp(w)) .[w]1/(p−1) Ap≤[w]1/2 Ap, p ∈[3,∞), In (8.4), the factor ([w]′ A∞)1/2comes from an estimate of Wilson [49] relating the weighted norms of the grand maximal function and a certain square function, while [w]1/2 Apis Lerner’s bound for the weighted norm of such square functions (whose exponent is optimal by [8]). To simplify comparison, let us only consider the simpler form of our bound (8.3). Then the sum of the powers of [w]Apand [w]′ A∞in both (8.3) and (8.4) is 2/p + (1 −2/p) = 1/2 + 1/2 = 1, and the sharper bound is the one where the larger weight constant [w]Aphas the smaller power. We have 2/p ≤1/2if and only if p≥4, and hence Lerner’s bound is sharper for p∈[3,4) and ours for p∈(4,∞). This indicates that the present results might not be the last word on joint Ap–A∞-control, but there is place for further investigation. 9. Proof of the end-point theory at p=∞ The proof again relies on the sharp reverse Hölder inequality Theorem 2.3: if w∈A∞and if we let r=r(w) := 1 + 1 cd[w]′ A∞ , then − ˆQ wrdx1 r ≤2 |Q|ˆQ w . 38 T. HYTÖNEN AND C. PÉREZ Proof of Theorem 1.26. For c=hfiQ, 1 w(Q)ˆQ|f−c|w=|Q| w(Q)− ˆQ|f−c|w ≤|Q| w(Q)− ˆQ|f−c|r(w)′1/r(w)′− ˆQ wr(w)1/r(w) ≤|Q| w(Q)Cdr(w)′kfkBMO2− ˆQ w =Cdr(w)′kfkBMO ≤Cd[w]′ A∞kfkBMO, which shows that kfkBMO(w)≤Cd[w]′ A∞kfkBMO. Note that we used the sharp order of growth of the local Lpnorms of BMO functions as p→ ∞, which follows easily from the exponential integrability. To see the sharpness for d= 1, consider w(x) = |x|−1+ε, which has [w]A∞h [w]′ A∞h1/ε and f(x) = log |x|. We check that kfkBMO(w)≥inf a 1 w([0,1]) ˆ1 0|log 1 x−a|w(x) dx≥c ε≥c[w]A∞≥c[w]′ A∞, which proves the claim. It is immediate that w([0,1]) = ´1 0x−1+εdx= 1/ε. It remains to compute ˆ1 0|log 1 x−a|x−1+εdx=ˆ∞ 0|t−a|e−εt dt=1 ε2ˆ∞ 0|u−εa|e−udu. It suffices to check that ψ(α) := ´∞ 0|u−α|e−udu≥c > 0for all α∈R. But this is an easy calculus exercise.  We now prove Corollary 1.27 on end-point estimates for Calderón–Zygmund operators. Proof of Corollary 1.27. For the positive estimate, it suffices to factorize T=I◦T, where T:L∞→BMO and I: BMO →BMO(w)have norm bounds cTand cd[w]′ A∞, respectively. Concerning sharpness, note that the Hilbert transform of χ(−1,0) is log(x+ 1) −log xfor x > 0. Since log(x+ 1) is bounded on [0,1], the computation proving the sharpness of the embedding BMO ֒→BMO(w)also gives the lower bound kHχ(−1,0)kBMO(|x|−1+ε)≥c/ε =c[x−1+ε]A∞kχ(−1,0)kL∞ ≥c[x−1+ε]′ A∞kχ(−1,0)kL∞. We conclude with the proof of Proposition 1.28 on the sharp relation of A∞and BMO. Note that here we use the larger constant [w]A∞, not [w]′ A∞. SHARP WEIGHTED BOUNDS INVOLVING A∞39 Proof of Proposition 1.28. Let Qbe a cube. We estimate ˆQ|log w−log c|=ˆQ∩{w≥c} log w c+ˆQ∩{w<c} log c w =ˆQ∩{w≥c} log w c+ˆQ−ˆQ∩{w≥c}log c w = 2 ˆQ∩{w≥c} log w c+ˆQ log c+ˆQ log 1 w ≤2ˆQ∩{w≥c} w c+|Q|log c+|Q|log [w]A∞.− ˆQ w. Hence − ˆQ|log w−log c| ≤ 2 c− ˆQ w+ log c+ log[w]A∞−log − ˆQ w. Choosing c=cQ= 2− ´Qw, we get − ˆQ|log w−log cQ| ≤ 1+log 2+log − ˆQ w+log[w]A∞−log − ˆQ w= log(2e[w]A∞), and this proves that klog wkBMO ≤log(2e[w]A∞). 9.1. Remark. In the last estimate, we cannot replace [w]A∞by [w]′ A∞. 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