Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces

Publ. Mat. 49 (2005), 73–91 EXTRAPOLATION AND SHARP NORM ESTIMATES FOR CLASSICAL OPERATORS ON WEIGHTED LEBESGUE SPACES Oliver Dragiˇ cevi´ c∗, Loukas Grafakos†, Mar´ ıa Cristina Pereyra‡and Stefanie Petermichl† Abstract We obtain sharp weighted Lpestimates in the Rubio de Francia extrapolation theorem in terms of the Apcharacteristic constant of the weight. Precisely, if for a given 1 <r<∞the norm of a sublinear operator on Lr(w) is bounded by a function of the Archaracteristic constant of the weight w, then for p > r it is bounded on Lp(v) by the same increasing function of the Apcharacteristic constant of v, and for p < r it is bounded on Lp(v) by the same increasing function of the r−1 p−1power of the Apcharacteristic constant of v. For some operators these bounds are sharp, but not always. In particular, we show that they are sharp for the Hilbert, Beurling, and martingale transforms. 1. Introduction 1.1. Extrapolation. A positive locally integrable function on Rnis called a weight. A weight wis said to be of class Ap, for 1 < p < ∞, if sup Q1 |Q|ZQ w 1 |Q|ZQ w−1 p−1p−1 <∞, 2000 Mathematics Subject Classification. 42A50, 42B20, 42B25, 46M35 (44A15, 47B38). Key words. Extrapolation, sharp weighted estimates, dyadic square function, dyadic paraproduct, martingale transform, Hilbert transform, Beurling transform. ∗Research supported by the European Commission (IHP network “Harmonic Analysis and Related Problems” 2002–2006, contract HPRN-CT-2001-00273-HARP). †Work supported by the NSF. ‡Research partially done while visiting the Centre de Recerca Matem`atica in Barcelona, Spain. 74 O. Dragiˇ cevi´ c et al. where the supremum is taken over all cubes Qin Rnwith sides parallel to the axes (Qwill always denote such cubes). The quantity above is called the Ap-characteristic constant of the weight wand will be denoted by kwkAp. A weight wis said to be of class A1if there is a constant C > 0 such that Mw ≤Cw a.e., where Mis the (uncentered) Hardy-Littlewood maximal function, i.e. Mf(x) = sup x∈Q 1 |Q|ZQ|f(y)|dy. The smallest possible Cis denoted by kwkA1. For an operator Tbounded from a Banach space Xinto itself (T∈ B(X)) we will denote by kTkXits operator norm. When 1 <q<∞, q0shall stand for the dual exponent of q, i.e. 1 q+1 q0= 1. Given a weight v on Rn,Lp(v) denotes the space of complex functions on Rnsuch that RRn|f|pvis finite. The following result is the celebrated extrapolation theorem of Rubio de Francia. Theorem (E). Assume we are given a sublinear1operator T:[ w∈Aq 1≤q<∞ Lq(w)−→ {all measurable complex-valued functions}. Suppose there is 1≤r < ∞such that T∈B(Lr(u)) for all weights u∈Ar, with bounds depending only on kukAr. Then T∈ B(Lp(w)) for all 1< p < ∞and all weights w∈Ap, with bounds depending only on kwkAp. More precisely, suppose for each B > 1there is a constant Nr(B)>0 such that we have (1) kTkLr(u)≤Nr(B)for all u∈Arwith kukAr≤B. Then for any 1< p < ∞and B > 1there is Np(B)>0such that for all weights w∈Apwith kwkAp≤B, (2) kTkLp(w)≤Np(B). 1It turns out that Tdoes not need to be sublinear, just well-defined on its domain, see [Gr, Section 9.5.b]. Extrapolation and Sharp Weighted Estimates 75 This result first appeared in [R]. Different proofs can be found in the books [GC-RF] and [Gr]. Muckenhoupt proved in [M] that for 1 <p<∞the maximal function is bounded on Lp(w) if and only if the weight wbelongs to the class Ap. Hunt, Muckenhoupt and Wheeden proved in [HMW] that the Apcondition also characterizes the boundedness of the Hilbert transform Hf(x) = p.v. 1 πZf(y) x−ydy in Lp(w). Coifman and Fefferman [CF] extended the theory to general Calder´on-Zygmund operators. In 1993, Buckley [Buc] obtained the following result concerning the Hardy-Littlewood maximal function2(1 < p < ∞): (3) kMkLp(w)≤C(p)kwkp0/p Ap, where the constant C(p) depends only on p(and the underlying dimension n). These bounds are sharp, i.e. kwkp0/p Apcannot be replaced by ϕ(kwkAp) for any function ϕ:R+→R+that grows slower than the p0/p-th power. This can be easily seen by using power functions and power weights. Taking w≡1 we see that the constants C(p) must blow up as p→1. In this note we use Buckley’s estimate (3) to improve Theorem (E) as follows. Theorem 1. With the notation and hypotheses as in Theorem (E), assume that Nr(B)denotes the smallest constant that satisfies inequality (1). Then for any 1<p<∞and all B > 1there is a constant Np(B) such that (2) holds for all weights win Apsatisfying kwkAp≤B. Moreover, Np(B)≤   21 rNr(2C(p0)p−r p−1B)if p > r 2r−1 rNr(2r−1(C(p)p−rB)r−1 p−1)if p < r. Here C(p)is the constant appearing in (3). 2Buckley actually obtained this result for the centered maximal function M0. However, the uncentered maximal function M, the centered one M0, and the dyadic maximal function Mdare comparable modulo dimensional constants, so (3) holds for either one. 76 O. Dragiˇ cevi´ c et al. This result, applied to the Beurling and martingale transforms for r= 2, N2(B) = CB and p > 2, was first observed in [PetV]. In this case, a careful extrapolation for p > 2 yields Np(B)≤CpB. That is, the linear dependence on the constant when p= 2 is preserved also for p > 2. However, this is not the case when p < 2, which motivates a more careful examination of the problem. 1.2. Sharp bounds. The linear bounds for the Beurling transform in Lp(w) in terms of kwkApfor p≥2 have important consequences in the theory of quasiconformal mappings. The connection is very well explained in the paper by Astala, Iwaniec and Saksman [AIS] who were interested in finding the minimal q < 2 for which all solutions to the Beltrami equation ¯ ∂f =µ·∂f that belong to the Sobolev space W1,q loc still self-improve to belonging to W1,2 loc , i.e. are quasiregular. Here µis a bounded function with kµk∞=k < 1. A deep result of Astala [A] says that q > 1 + k suffices. On the other hand, Iwaniec and Martin [IM] found examples showing that the result could in general not be true for q < 1 + k. In [AIS] the borderline case q= 1 + kwas addressed; it was pointed out by the authors that quasiregularity would be a consequence of the linear dependence of the norm of the Beurling transform on weighted spaces Lp(w) for p≥2 in terms of the Apcharacteristic of the weight w. This linear dependence was settled in [PetV] and later in [DV] for p≥2, the only range for which it is true. For the maximal function, the bound for kMkL2(w)is also linear in kwkA2, see (3). If 1 <p<2, extrapolation yields sharp dependence of kMkLp(w)on kwkAp. However, for p > 2, extrapolation only gives linear growth on kwkAp, when the sharp growth is kwkp0/p Ap. In [Buc], Buckley considers two more examples were the same phenomena occur. He shows that a parametric class of Marcinkiewicz integral operators is uniformly bounded on Lp(w) by kwkApfor all 1 < p < ∞and these linear estimates are sharp [Buc, Theorem 2.15]. In particular, extrapolating from the sharp linear estimate at p= 2 yields the right sharp linear estimate for p > 2, but for p < 2 it yields a worse estimate. Buckley also shows that a parametric class of averaging operators is uniformly bounded on Lp(w)bykwk1/p Apfor all 1 < p < ∞[Buc, Lemma 2.18]. In this case, starting from the estimates on Lr(w) for any 1 < r < ∞, extrapolation yields an estimate that is worse than the sharp estimate for all p6=r. Therefore the estimates of Theorem 1 may not be sharp for some operators even when the initial estimate is sharp. However, the Extrapolation and Sharp Weighted Estimates 77 theorem itself is sharp, as we will show that for a variety of classical operators that have a sharp linear norm estimate in L2(w), the extrapolated bounds are also sharp for all 1 < p < ∞. Buckley [Buc] also showed that the Hilbert transform —and for that matter convolution singular integral operators with Calder´on-Zygmund kernels— are bounded on Lp(w) with an operator norm which is at most a multiple of kwkα Ap, where max{1, p0/p} ≤ α≤p0. In particular, for p= 2 he showed that the dependence on kwkA2was at least linear, and at most quadratic. Recently there has been renewed interest in computing the exact dependence of the operator norms on the Apcharacteristic constant of the weight. Sharp linear dependence on kwkA2was obtained by Hukovic, Treil, and Volberg [Huk], [HukTV] for the dyadic square function on L2(w) and for the martingale transform, a dyadic model for singular integral operators, by Wittwer [W1], [W2]. As we already mentioned, analogous results were recently obtained for the Beurling transform by Petermichl and Volberg [PetV], and later by Dragiˇcevi´c and Volberg [DV]. Petermichl and Pott [PetPot] very elegantly showed that α≤3/2 for the Hilbert transform. Petermichl [Pet] improved this estimate to α= 1 when p≥2. The difficulty in [Pet] was to obtain linear dependence of kHkL2(w)on kwkA2; extrapolation then gave the same dependence for p > 2. Using Theorem 1 we obtain that the norms of these operators on Lp(w) are bounded by at most a multiple of kwkα Ap, α= max{1, p0/p}, for all 1 < p < ∞. As mentioned earlier, using power weights and power functions, Buckley [Buc] showed that for convolution operators with Calder´on-Zygmund kernels the power is at least max{1, p0/p}. Hence in the cases of Hilbert and Beurling transform, Theorem 1 provides the sharp bounds. If we could prove linear bounds for all convolution operators with CZ-kernels, then by extrapolation we will obtain the same sharp bounds in Lp(w) as for the Hilbert and the Beurling transform. Obtaining the linear bounds in L2(w) can be very difficult. For instance, it is not yet known to the authors whether there is a bound for the first-order Riesz transforms on L2(w) depending linearly on kwkA2. We will show that the bounds obtained by extrapolation for the martingale transform are also sharp for all 1 < p < ∞. We can show that the extrapolated bounds for the square function are sharp for p < 2. It is not clear yet that the linear bound obtained by extrapolation for p > 2 is sharp, so far we can show that it must be at least of the order kwkp0/p Ap. We can summarize all these results in the following theorem. 78 O. Dragiˇ cevi´ c et al. Theorem 2. Let Tbe any of the Hilbert transform, the Beurling transform, the martingale transform, or the dyadic square function. Then for any 1< p < ∞there exist positive constants Cpsuch that for all weights win Apwe have (4) kTkLp(w)≤Cpkwkα Ap, where α= max{1, p0/p}. The exponent αin this estimate is sharp for the Hilbert, Beurling and martingale transforms for all 1<p<∞. For the dyadic square function the exponent is sharp for 1< p ≤2. All results establishing the linear bounds for the above operators on L2(w) have been obtained using the technique of Bellman functions introduced by Nazarov, Treil and Volberg [NTV] in the harmonic analysis context; see [NT] for an extensive introduction to this technique. The linear upper bound in Theorem 2 for p > 2 was previously known for the martingale, Hilbert and Beurling transforms. Unfortunately, extrapolation does not preserve the nature of the initial estimate on Lr(w) for all 1 < p < ∞, only for p > r. Therefore sharpness at the given rdoes not automatically transfer to all other p∈(1,∞). One has to check sharpness by other means for each p6=r. In all the examples discussed we search for a function and a weight (or a family of functions and weights) that will provide a lower bound estimate of the same order of the upper bound, therefore showing that the estimate is indeed sharp. Acknowledgement. The authors would like to thank the referee for some very useful suggestions that improved the presentation. 2. Some Lemmata The first two lemmata below correspond to IV.5.16 and IV.5.17 in [GC-RF]. The case r= 2 and p > 2 was carefully calculated in [PetV]. Lemma 1. Take p, s > 1,w∈Apand u∈Ls(w). Let (5) S(u) = w−1M(|u|s/p0w)p0/s . (a) Then Sis bounded in Ls(w), moreover, kSkLs(w)≤C(p0)p0/skwkp0/s Ap. (b) Let p,sbe such that r:= p/s0∈[1,∞). Take a nonnegative function u∈Ls(w). Extrapolation and Sharp Weighted Estimates 79 If r > 1, then the pair (uw, S(u)w)belongs to the class Ar. Furthermore, sup Q1 |Q|ZQ uw 1 |Q|ZQ (S(u)w)−1 r−1r−1 ≤ kwk1−p0 s Ap. If r= 1, the A1condition on the pair (uw, S(u)w)also holds and translates into M(uw)≤S(u)w. Proof: (a) Estimating directly the norm we obtain kSukLs(w)=Zhw−1M(|u|s/p0w)ip0 w1 s =ZhM(|u|s/p0w)ip0 w1−p01 s ≤ kMkp0/s Lp0(w1−p0)k|u|s/p0 wkp0/s Lp0(w1−p0)=kMkp0/s Lp0(w1−p0) kukLs(w). It only remains to insert Buckley’s sharp (3) estimate and to recall that w∈Apimplies w1−p0∈Ap0, moreover (6) kw1−p0kAp0=kwk 1 p−1 Ap=kwkp0/p Ap. All together, these facts imply kMkLp0(w1−p0)≤C(p0)kw1−p0k(p0)0/p0 Ap0=C(p0)(kwkp0/p Ap)p/p0=C(p0)kwkAp. Thus, kSkLs(w)≤C(p0)p0/skwkp0/s Ap, as claimed. (b) If s=p0, we have r= 1, then S(u)w=M(uw). Automatically the two-weight A1condition, M(uw)≤S(u)w, holds. If s > p0>1, then p > s0>1 and r > 1. Note that (r−1) = (p−1) 1−p0 sand, by definition of the maximal function, Dus/p0wEQ≤sup x∈Q M(us/p0w)(x). 80 O. Dragiˇ cevi´ c et al. Here hfiQdenotes the mean of the function fover the cube Q. Consequently, huwiQD[S(u)w] −1 r−1Er−1 Q=huwiQD[(w−1M(us/p0w))p0/sw] −1 r−1Er−1 Q =Du wp0/sw1−p0/sEQ D[M(us/p0w)]p0 s −1 r−1w −1 p−1Er−1 Q ≤Dus/p0 wEp0/s Qhwi1−p0 s QDus/p0 wE−p0 s QDw −1 p−1E(p−1) “1−p0 s” Q =hwiQDw −1 p−1Ep−1 Q1−p0 s ≤kwk1−p0 s Ap. Taking supremum on the left-hand-side, over all cubes Qwith sides parallel to the axis, we obtain the desired inequality. Lemma 2. Let p,s,rand wbe as in the previous lemma. Then for each u≥0,u∈Ls(w), there exists v∈Ls(w)such that (a) u(x)≤v(x)a.e. and kvkLs(w)≤2kukLs(w). (b) vw ∈Ar, moreover, kvwkAr≤2C(p0)p0/skwkAp. Proof: Define vvia the following convergent Neumann series: v= ∞ X n=0 Sn(u) 2nkSkn=u+S(u) 2kSk+···, where kSk=kSkLs(w). Then (a) is clearly satisfied. (b) It follows from the definition of vand the sublinearity of Sthat Sv ≤2kSk(v−u)≤2kSkv. Suppose r > 1. By the previous lemma, the pair (vw, S(v)w) lies in Ar with its Ar-constant bounded by kwk1−p0 s Ap. Also recall that kSk ≤ C(p0)p0/skwkp0/s Ap. We can now estimate kvwkAr: hvwiQD(vw) −1 r−1Er−1 Q≤ hvwiQD(S(v)w) −1 r−1Er−1 Q2kSk ≤ kwk1−p0 s Ap2C(p0)p0/skwkp0/s Ap= 2C(p0)p0/skwkAp. Extrapolation and Sharp Weighted Estimates 81 Taking supremum on the left hand side, over all cubes Qwith sides parallel to the axis, we obtain the desired estimate for kvwkAr,r > 1. When r= 1, then s=p0and kSk ≤ C(p0)kwkAp, furthermore M(vw)≤S(v)w≤2kSkvw ≤2C(p0)kwkApvw. We conclude that kvwkA1≤2C(p0)kwkAp, as claimed. The next lemma appears as IV.5.18 in [GC-RF]; see also Lemma 9.5.4 in [Gr] for a slightly different method for part (b) which yields the same bounds as here. Attention was paid to the constants in [Gr] but Buckley’s sharp estimate for kMkLp(w)was missing; with this additional information, the constants in [Gr] would be of the same order as the ones obtained here. Lemma 3. Fix rsatisfying 1≤r < ∞. (a) Let 1≤r < p < ∞and s= (p/r)0. Let w∈Ap, then for every u≥0,u∈Ls(w), there exists v≥0,v∈Ls(w), such that u(x)≤v(x)and kvkLs(w)≤2kukLs(w). Moreover, vw ∈Arand kvwkAr≤2C(p0)p−r p−1kwkAp. (b) Let 1< p < r and s=p r−p. Let w∈Ap, then for every u≥0, u∈Ls(w), there exists v≥0,v∈Ls(w)such that, u(x)≤v(x), and kvkLs(w)≤2r−1kukLs(w). Moreover, v−1w∈Arand kv−1wkAr≤2r−1(C(p)r−pkwkAp)r−1 p−1. Here C(p)denotes the constant in (3). Proof: (a) Clearly r≥1 implies s0≤p, and we can now use Lemma 2 after observing that p0 s=p−r p−1. (b) Take p,rand sas in the formulation of the lemma. (Notice that everything that is being said still holds if 0 <s<1.) Now the dual exponents satisfy the opposite inequality, r0< p0, and if we define s∗:= (p0/r0)0>1, then s∗=s(r−1). We apply the previous case with p0,r0and w1−p0∈Ap0instead of p,r and w∈Ap, respectively. If u≥0, u∈Ls(w), then u0=us/s∗wp0/s∗∈ Ls∗(w1−p0) and by (a) there exists v0∈Ls∗(w1−p0) such that u0≤v0a.e.,kv0kLs∗(w1−p0)≤2ku0kLs∗(w1−p0),and v0w1−p0∈Ar0,kv0w1−p0kAr0≤2C(p)p0−r0 p0−1kw1−p0kAp0 = 2C(p)r−p p−1kwk 1 p−1 Ap. 88 O. Dragiˇ cevi´ c et al. So far this was true for any σ. Now choose σIk= (−1)k. We get kTσfkp Lp(w)≥1 δp ∞ X n=1  (−1)n+1 2n+1 −X k>n+1 (−1)k 2k p 2n(p+δ−pδ) =2p (3δ)pX n≥1 2−n(p−1)δ=2p (3δ)p(2(p−1)δ−1) ∼1 δp+1(p−1) ∼C0(p)kwkp0 Apkfkp Lp(w). Taking p-th roots we conclude that sup σkTσkLp(w)≥C00(p)kwkp0/p Ap. Thus ϕ(x) = xp0/p is sharp for p < 2, and by duality ϕ(x) = xis sharp for p > 2. 4.3. The dyadic paraproduct. A locally integrable function bis said to be in dyadic BMOd, if the average oscillation of bis uniformly bounded on dyadic intervals. More precisely, if kbkBMOd= sup J∈D 1 |J|ZJ|b(x)−hbiI|dx < ∞. For each function b∈BMOdthe dyadic paraproduct πbis defined by πbf(x) = X I∈D hfiIhb, hIihI(x). It is known that the dyadic paraproduct is bounded in Lp(w) whenever w∈Ap; see [KPer]. The following quadratic estimate can be shown to hold [PerPet] kπbfkL2(w)≤K(kbkBMO )kwk2 A2kfkL2(w). (We do not think this is sharp, we believe the sharp estimate should be linear as for all other operators studied in this paper.) Theorem 1 then gives the upper bound kπbfkLp(w)≤Kp(kbkBMO )kwk2α ApkfkLp(w), where α= max{1, p0/p}. Extrapolation and Sharp Weighted Estimates 89 References [A] K. Astala, Area distortion of quasiconformal mappings, Acta Math. 173(1) (1994), 37–60. [AIS] K. Astala, T. Iwaniec and E. Saksman, Beltrami operators in the plane, Duke Math. J. 107(1) (2001), 27–56. [Buc] S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340(1) (1993), 253–272. [CF] R. R. Coifman and C. 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Extrapolation and Sharp Weighted Estimates 91 Oliver Dragiˇcevi´c: Scuola Normale Superiore Piazza dei Cavalieri 7 56126 Pisa Italy E-mail address:[email protected] Loukas Grafakos: Department of Mathematics University of Missouri Columbia, MO 65211 USA E-mail address:[email protected] Mar´ıa Cristina Pereyra: Department of Mathematics and Statisitics University of New Mexico Albuquerque, NM 87131 USA E-mail address:[email protected] Stefanie Petermichl: Department of Mathematics Brown University Box 1917 Providence, RI 02912 USA E-mail address:[email protected] Primera versi´o rebuda el 16 de desembre de 2003, darrera versi´o rebuda el 26 d’abril de 2004.