We give a sufficient condition for singular integral operators and, more
generally, Calder´on-Zygmund operators to satisfy the weak (p, p) inequality
u({x ∈ R n : |T f(x)| > t}) ≤ C / tp Z Rn |f|p v dx, 1 < p < ∞. Our condition is an Ap-type condition in the scale of Orlicz spaces: kukL(log L) p−1+δ,Q 1 |Q|
Z Q v −p 0/p dx p/p0 ≤ K 0. This conditions is stronger than the Ap condition and is sharp since it fails when δ = 0.Dirección General de Investigación Científica y Técnic

Cruz Uribe, David

Pérez Moreno, Carlos

idUS. Depósito de Investigación Universidad de Sevilla

Mathematical Research Letters, 6 1999, 1-11.SHARP TWO-WEIGHT, WEAK-TYPE NORM INEQUALITIESFOR SINGULAR INTEGRAL OPERATORSD. CRUZ-URIBE, SFO AND C. PE´REZAbstract. We give a sufficient condition for singular integral operators and, moregenerally, Caldero´n-Zygmund operators to satisfy the weak (p, p) inequalityu({x ∈ Rn : |Tf(x)| > t}) ≤ Ctp∫Rn|f |pv dx, 1 0.This conditions is stronger than the Ap condition and is sharp since it fails whenδ = 0.1. IntroductionLet T be a Caldero´n-Zygmund singular integral operator:Tf(x) = p.v.∫RnK(x− y)f(y) dy,where the kernel K is continuously differentiable on Rn \ {0}, has zero average onthe unit sphere, and for all x 6= 0,|K(x)| ≤ C|x|n and |∇K(x)| ≤C|x|n+1 .More generally, we may assume that T is a Caldero´n-Zygmund operator. For aprecise definition, see Coifman and Meyer [4], Journe´ [9] or Christ [3]. In this paperwe consider the problem of finding sufficient conditions on a pair of weights (u, v)—i.e. non-negative, locally integrable functions—such that T satisfies the weak (p, p)1991 Mathematics Subject Classification. 42B20,42B25.Key words and phrases. weights, singular integral operators, Caldero´n-Zygmund operators, max-imal operators, Orlicz spaces.The first author is supported by a Ford Foundation fellowship; the second author by DGICYTGrant PB40192, Spain. This work was finished while the second author enjoyed the hospitality ofthe Centre de Recerca Matematica, Barcelona.12 D. CRUZ-URIBE, SFO AND C. PE´REZinequality (1 t}) ≤ Ctp∫Rn|f |pv dx, t > 0, f ∈ Lp(v).In the one-weight case—i.e. u = v—a sufficient condition is v ∈ Ap, and this conditionis necessary if n = 1 and T is the Hilbert transform. (For sufficiency, see Journe´ [9];for necessity, see Hunt, Muckenhoupt and Wheeden [8].) In the two-weight case,however, (u, v) ∈ Ap is no longer sufficient: see Muckenhoupt and Wheeden [12]or section 4 below for a counter-example. Recall that (u, v) ∈ Ap if there exists apositive constant K such that for all cubes Q,(1|Q|∫Qu dx)(1|Q|∫Qv−p′/p dx)p/p′≤ K.Neugebauer [13] showed that if there exists r > 1 such that (ur, vr) ∈ Ap then (1.1)holds, and in fact T is strong (p, p). In [5] we showed that if we strengthened the Apcondition by only adding a “power bump” to the left-hand term, then we obtained asufficient condition for T to be weak (p, p).Theorem 1.1. Let T be a Caldero´n-Zygmund operator. Given a pair of weights(u, v) and p, 1 1 and for all cubes Q,(1.2)(1|Q|∫Qur dx)1/r (1|Q|∫Qv−p′/p dx)p/p′≤ K <∞.Then inequality (1.1) holds.In this paper we improve Theorem 1.1 by replacing the power bump in (1.2) by asmaller “Orlicz bump”. Given a Young function B and a cube Q, the mean Luxem-bourg norm of a measurable function f on Q is defined by(1.3) ‖f‖B,Q = inf{λ > 0 :1|Q|∫QB( |f |λ)dx ≤ 1}.When B(t) = t log(1 + t)p−1+δ, this norm is also denoted by ‖ · ‖L(logL)p−1+δ ,Q. (Moreinformation on Orlicz spaces is in Section 2 below.) Our main result is the following.Theorem 1.2. Let T be a Caldero´n-Zygmund operator. Given a pair of weights(u, v) and p, 1 0 and for all cubes Q,(1.4) ‖u‖L(logL)p−1+δ ,Q(1|Q|∫Qv−p′/p dx)p/p′≤ K <∞.TWO-WEIGHT NORM INEQUALITIES 3Then for every t > 0 and f ∈ Lp(v),(1.5) u({x ∈ Rn : |Tf(x)| > t}) ≤ Ctp∫Rn|f |pv dx.Further, this result is sharp since it does not hold in general when δ = 0.Remark 1.3. Theorem 1.2 is stronger than the result proved in [5], but our proof onlyworks for Caldero´n-Zygmund operators. We do not know if the corresponding resultsproved in [5] for fractional integrals and commutators are true with power bumpsreplaced by Orlicz bumps.Remark 1.4. As a corollary to Theorem 1.2 we get a new proof of the one-weight,strong (p, p) norm inequality for Caldero´n-Zygmund operators. If w ∈ Ap then bothw and w−p′/p satisfy the reverse Ho¨lder inequality, so for some  > 0, (1.4) holds withp replaced by p± . The strong-type inequality then follows by interpolation.Remark 1.5. An analogue of Theorem 1.2 for the two-weight, strong (p, p) inequalityfor the Hilbert transform on the unit circle was proved by Treil, Volberg and Zheng[21]. However, their method does not extend to Caldero´n-Zygmund operators.The proof of Theorem 1.2 proceeds as follows: given a Young function A, define themaximal functionMAf(x) = supQ3x‖f‖A,Q.In [16] it was shown that if A(t) = t log(1+t),  > 0, and T is any Caldero´n-Zygmundoperator, then for any weight w and function f ,(1.6) w({x ∈ Rn : |Tf(x)| > t}) ≤ Ct∫Rn|f |MAw dx.We combine this inequality with a duality argument to reduce the proof to charac-terizing the weights (u, v) such that MA is bounded from Lp′(u−p′/p) to Lp′(v−p′/p).We do this by extending the techniques used in [18].We note that independently and much earlier, Muckenhoupt and Wheeden [11] con-jectured that inequality (1.6) holds when T is the Hilbert transform and MA isreplaced by the Hardy-Littlewood maximal operator. Assuming this, they provedthat inequality (1.5) holds for the Hilbert transform and for pairs (u, v) such that(u−p′/p, v−p′/p) satisfy the Sawyer condition Sp′ . (They also conjectured that thiscondition is necessary.) Their proof uses a duality argument which is essentially thesame as ours.If the Muckenhoupt-Wheeden conjecture is true for Caldero´n-Zygmund operators,then our proof of Theorem 1.2 yields the stronger result that inequality (1.5) holdsif in inequality (1.4) we replace the Orlicz norm ‖ · ‖L(logL)p−1+δ ,Q by the weaker4 D. CRUZ-URIBE, SFO AND C. PE´REZOrlicz norm ‖ ·‖B,Q, where B is a doubling Young function which satisfies the growthcondition(1.7)∫ ∞c(tB(t))p′−1dtt<∞.For example, we can take B(t) ≈ t log(t) log log(t)p−1+δ, δ > 0. (See Section 2 for therelevance of this condition.) Irrespective of the validity of the Muckenhoupt-Wheedenconjecture, we believe this result is true.The remainder of this paper is organized as follows: in Section 2 we gather a fewresults about Orlicz spaces (some of which we alluded to above) that we will usein the proofs. In Section 3 we prove a sufficient, Ap-type condition in the scale ofOrlicz spaces for the strong (p, p) norm inequality for MA. Since we can do so withno additional work, we prove the result for a larger class of Young functions A thanis necessary to prove Theorem 1.2. Finally, in Section 4 we prove Theorem 1.2. Ourproof actually gives a slightly stronger result; we state it precisely at the end of thesection.Throughout this paper all notation is standard or will be defined as needed. Allcubes are assumed to have their sides parallel to the co-ordinate axes. Given a cubeQ, l(Q) will denote the length of its sides and for any r > 0, rQ will denote the cubewith the same center as Q and such that l(rQ) = rl(Q). By weights we will alwaysmean non-negative, locally integrable functions which are positive on a set of positivemeasure. Given a Lebesgue measurable set E and a weight w, |E| will denote theLebesgue measure of E and w(E) =∫Ew dx. Given 1 < p < ∞, p′ = p/(p − 1) willdenote the conjugate exponent of p. Finally, C will denote a positive constant whosevalue may change at each appearance.2. Preliminary Results on Orlicz SpacesIn this section we summarize a few facts about Orlicz spaces. (For more information,see Bennett and Sharpley [1] or Rao and Ren [19].) A function B : [0,∞) → [0,∞)is a doubling Young function if it is continuous, convex and increasing, if B(0) = 0and B(t) → ∞ as t → ∞, and if it satisfies B(2t) ≤ CB(t) for all t > 0. For Orlicznorms we are usually only concerned about the behavior of Young functions for tlarge. Given two functions B and C, we write B(t) ≈ C(t) if B(t)/C(t) is boundedand bounded below for t ≥ c > 0.Recall that we defined the localized Luxembourg norm by equation (1.3); an equiv-alent norm which is often useful in calculations is due to Krasnosel’ski˘ı and Ruticki˘ıTWO-WEIGHT NORM INEQUALITIES 5[10, p. 92] (also see Rao and Ren [19, p. 69]):(2.1) ‖f‖A,Q ≤ infµ>0{µ+µ|Q|∫QA( |f |µ)dx}≤ 2‖f‖A,Q.A Young function B is said to satisfy the Bp condition if for some constant c > 0,∫ ∞cB(t)tpdtt≈∫ ∞c(tp′B¯(t))p−1dtt<∞.(Cf. condition (1.7) above.) Here B¯ is the complementary Young function associatedto B; it has the property that for all t > 0,t ≤ B−1(t)B¯−1(t) ≤ 2t.In [18] it was shown that if B is doubling then B ∈ Bp if and only if MB : Lp → Lp isbounded. Examples of functions satisfying the Bp condition are B(t) = tq, 1 ≤ q < p.More interesting examples are given byB(t) = tp log(1 + t)−(1+δ) B(t) ≈ tp log(t)−1 log log(t)−(1+δ), δ > 0.Finally, we will need a generalization of Ho¨lder’s inequality due to O’Neil [14]. (Alsosee Rao and Ren [19, p. 64].)Lemma 2.1. Let A, B and C be Young functions such that for all t > 0,B−1(t)C−1(t) ≤ A−1(t).Then for all functions f and g and all cubes Q,‖fg‖A,Q ≤ 2‖f‖B,Q‖g‖C,Q.3. Strong-type norm inequalities for MAIn this Section we prove the following weighted norm inequality.Theorem 3.1. Given p, 1 0, and C is doubling and satisfies the Bp condition. If(u, v) is a pair of weights such that for every cube Q,(3.1)(1|Q|∫Qu dx)1/p‖v−1/p‖B,Q ≤ K <∞,then for every function f ∈ Lp(v),∫Rn(MAf)pu dx ≤ C∫Rn|f |pv dx.6 D. CRUZ-URIBE, SFO AND C. PE´REZRemark 3.2. Since B−1(t)→∞ as t→∞, C−1(t) ≤ A−1(t) for all t sufficiently large.Hence A(t) ≤ C(t) for t large and so A ∈ Bp. As we noted above, when u = v = 1this condition is also necessary.We first prove a lemma which generalizes the Caldero´n-Zygmund decomposition.Lemma 3.3. Given a Young function A, suppose f is a non-negative function suchthat ‖f‖A,Q tends to zero as l(Q) tends to infinity. Given a > 2n+1, for each k ∈ Zthere exists a disjoint collection of maximal dyadic cubes {Qkj} such that for each j,(3.2) ak < ‖f‖A,Qkj ≤ 2nak,and{x ∈ Rn : MAf(x) > 4nak} ⊂⋃j3Qkj .Further, let Dk =⋃j Qkj and Ekj = Qkj \ (Qkj ∩ Dk+1). Then the Ekj ’s are pairwisedisjoint for all j and k and there exists a constant α > 1, depending only on a, suchthat |Qkj | ≤ α|Ekj |.Remark 3.4. To have f satisfy the hypotheses of Lemma 3.3, it suffices to assumethat f is bounded and has compact support.Proof. The existence of dyadic cubes {Qkj} with the desired properties is shown in[18]. In the case A(t) = t the existence of α is proved in [17] (cf. Garcia-Cuerva andRubio de Francia [7, p. 398]). Here we extend that proof to general A.Since the Qkj ’s are disjoint for fixed k, by their maximality the Ekj ’s are pairwisedisjoint for all j and k. Further, by inequality (3.2),|Qkj ∩Dk+1| =∑i|Qkj ∩Qk+1i |=∑i:Qk+1i ⊂Qkj|Qk+1i |≤a−k−1∑i:Qk+1i ⊂Qkj|Qk+1i |‖f‖A,Qk+1i .TWO-WEIGHT NORM INEQUALITIES 7By inequality (2.1),≤a−k−1∑i:Qk+1i ⊂Qkj|Qk+1i | infµ>0{µ+µ|Qk+1i |∫Qk+1iA( |f |µ)dx}=a−k−1∑i:Qk+1i ⊂Qkjinfµ>0{µ|Qk+1i |+ µ∫Qk+1iA( |f |µ)dx}≤a−k−1 infµ>0{µ|Qkj ∩Dk+1|+ µ∫Qkj∩Dk+1A( |f |µ)dx}≤a−k−1 infµ>0{µ|Qkj |+ µ∫QkjA( |f |µ)dx}≤2a−k−1|Qkj |‖f‖A,Qkj .Again by inequality (3.2),≤2n+1a−1|Qkj |.It follows immediately from this that|Ekj | ≥ |Qkj | − 2n+1a−1|Qkj |,which establishes the desired inequality with α = (1− 2n+1a−1)−1. Proof of Theorem 3.1. Fix a function f ∈ Lp(v). Without loss of generality we mayassume that f is non-negative; by a standard approximation argument we may alsoassume that f is bounded and has compact support.Fix a > 2n+1; for k ∈ Z letΩk = {x ∈ Rn : 4nak < MAf(x) ≤ 4nak+1}.Then by Lemma 3.3,Ωk ⊂⋃j3Qkj , where ‖f‖A,Qkj > ak.8 D. CRUZ-URIBE, SFO AND C. PE´REZWe can therefore estimate as follows:∫Rn(MAf)pu dx =∑k∫Ωk(MAf)pu dx≤C∑kakpu(Ωk)≤C∑j,kakpu(3Qkj )≤C∑j,ku(3Qkj )‖f‖pA,Qkj .By Lemma 2.1,=C∑j,ku(3Qkj )‖fv1/pv−1/p‖pA,Qkj≤C∑j,ku(3Qkj )‖v−1/p‖pB,Qkj ‖fv1/p‖pC,Qkj.It follows from (1.3) that ‖v−1/p‖B,Qkj ≤ 3n‖v−1/p‖B,3Qkj . Hence, by Lemma 3.3 andcondition (3.1), and since the Ekj ’s are disjoint,≤C∑j,k(1|3Qkj |∫3Qkju dx)‖v−1/p‖pB,3Qkj‖fv1/p‖pC,Qkj|Ekj |≤∑j,k∫EkjMC(fv1/p)p dx≤C∫RnMC(fv1/p)p dx.≤C∫Rnfpv dx.The last inequality holds since C is doubling and in Bp, and so as we noted in Section2, MC is bounded on Lp. This completes the proof. TWO-WEIGHT NORM INEQUALITIES 94. Proof of Theorem 1.2In this section we prove Theorem 1.2. Fix p, 1 0. Let A(t) =t log(1 + t), where 0 <  < δ/p. Let η = δ − p. ThenA−1(t) ≈ tlog(1 + t)=t1/plog(1 + t)+(p−1+η)/p× t1/p′ log(1 + t)(p−1+η)/p=B−1(t)C−1(t),whereB(t) ≈ tp log(1 + t)(1+)p−1+η = tp log(1 + t)p−1+δandC(t) ≈ tp′ log(1 + t)−1−(p′−1)η.(This triple of Young functions is originally due to O’Neil [15].) It follows at oncefrom the definition that C is doubling and satisfies the Bp′ condition. Therefore, byTheorem 3.1, MA : Lp′(u−p′/p)→ Lp′(v−p′/p) is bounded, provided that(1|Q|∫Qv−p′/p dx)1/p′‖u1/p‖B,Q ≤ K <∞.But this is equivalent to condition (1.4), since by a change of variables, ‖u1/p‖B,Q ≈‖u‖1/pL(logL)p−1+δ .Now fix f . By a standard approximation argument we may assume without loss ofgenerality that f ∈ C∞(Rn) and has compact support.For each t > 0, let Ωt = {x ∈ Rn : |Tf(x)| > t}. This set is bounded, so u(Ωt) <∞.By duality, there exists a non-negative function G ∈ Lp′ , ‖G‖p′ = 1, such thatu(Ωt)1/p =‖u1/pχΩt‖p=∫Ωtu1/pGdx.By inequality (1.6),≤Ct∫Rn|f |MA(u1/pG) dx;10 D. CRUZ-URIBE, SFO AND C. PE´REZby Ho¨lder’s inequality,=Ct∫Rn|f |v1/pv−1/pMA(u1/pG) dx≤C(1tp∫Rn|f |pv dx)1/p(∫RnMA(u1/pG)p′v−p′/p dx)1/p′.By the boundedness of MA noted above, and since ‖G‖p′ = 1,≤C(1tp∫Rn|f |pv dx)1/p(∫RnGp′dx)1/p′=C(1tp∫Rn|f |pv dx)1/p.If we raise both sides of the final inequality to the power p we get inequality (1.5).To show that the hypotheses of Theorem 1.2 are sharp, consider the following exampleon the real line due to Muckenhoupt [11]. Fix p = 2; it follows from an inequalitydue to Stein [20] that for any function u, ML(logL)u ≈ M2u. (Also see Carozza andPassarelli [2].) Further, for any cube Q,‖u‖B,Q · 1|Q|∫QMBu(x)−1 dx ≤ 1|Q|∫QMBu(x)MBu(x)−1 dx ≤ 1,so the pair (u,MBu) satisfies inequality (1.4) with δ = 0. If we letu(x) = (x log x log log x)−1 χ[ee,∞),then a lengthy but straight-forward calculation shows that on [eee,∞),M2u(x) ≈ x−1 log x log log log x.Letf(x) = (log x log log x)−1 χ[eee ,∞);then ∫Rn|f |2MBu dx =∫ ∞eeelog log log xx log x(log log x)2dx <∞.On the other hand, for x > ee, Hf(x) = −∞, sou({x ∈ R : |Hf(x)| > 1}) ≥∫ ∞eeu dx =∞.Therefore, inequality (1.5) does not hold. Final Remark: Our proof of Theorem 1.2 yields a somewhat more general result.Theorem 3.1 clearly has weaker hypotheses, and though not so stated, the proof ofTWO-WEIGHT NORM INEQUALITIES 11inequality (1.6) in [16] shows that it holds for larger class of Young functions. If wecombine hypotheses we get the following result whose details are left to the reader.Theorem 4.1. Let T be a Caldero´n-Zygmund operator. Given p, 1 0, B−1(t)C−1(t) ≤ A−1(t);(2) A is doubling and there exists q > 1 such that∫ ∞c(tA(t))q−1dtt<∞;(3) C is doubling and satisfies the Bp′ condition.Then, if (u, v) is a pair of weights such that for every cube Q,‖u1/p‖B,Q(1|Q|∫Qv−p′/p dx)1/p′≤ K <∞,inequality (1.5) holds for all f ∈ Lp(v).References[1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988.[2] M. Carozza and A. Passarelli di Napoli, Composition of maximal operators, Publ. Mat. 40(1996), 397-409.[3] M. Christ, Lectures on Singular Integral Operators, CBMS Regional Conference Series in Math-ematics, 77, Amer. Math. Soc., Providence, 1990.[4] R. Coifman and Y. Meyer, Au dela` des ope´rateurs pseudo-diffe´rentiels, Aste´risque 57 (1979),1-84.[5] D. Cruz-Uribe, SFO, and C. Pe´rez, Two-weight, weak-type norm inequalities for fractionalintegrals, Caldero´n-Zygmund operators and commutators, preprint.[6] C. Fefferman and E.M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115.[7] J. Garc´ıa-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics,North Holland Math. Studies 116, North Holland, Amsterdam, 1985.[8] R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugatefunction and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-252.[9] J.-L. Journe´, Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral ofCaldero´n, Lecture Notes in Mathematics, 994, Springer Verlag, New York, 1983.[10] M.A. Krasnosel’ski˘ı and Ya.B. Ruticki˘ı, Convex functions and Orlicz spaces, P. Noordhoff,Groningen, 1961.[11] B. Muckenhoupt, private communication.[12] B. Muckenhoupt and R. Wheeden, Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform, Studia Math. 60 (1976), 279-294.[13] C.J. Neugebauer, Inserting Ap-weights, Proc. Amer. Math. Soc. 87 (1983), 644-648.[14] R. O’Neil, Fractional integration in Orlicz spaces, Trans. Amer. Math. Soc. 115 (1965), 300-328.[15] R. O’Neil, Integral transforms and tensor products on Orlicz spaces and Lp,q spaces, J. D’Anal.Math. 21 (1968), 1-276.12 D. CRUZ-URIBE, SFO AND C. PE´REZ[16] C. Pe´rez, Weighted norm inequalities for singular integral operators, J. London Math. Soc. 49(1994), 296-308.[17] C. Pe´rez, Two weighted inequalities for potential and fractional type maximal operators, IndianaMath. J. 43 (1994), 663-683.[18] C. Pe´rez, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operatorbetween weighted Lp-spaces with different weights, Proc. London Math. Soc. 71 (1995), 135-57.[19] M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.[20] E.M. Stein, Note on the class L logL, Studia Math. 32 (1969), 305-310.[21] S. Treil, A. Volberg and D. Zheng, Hilbert Transform, Toeplitz operators and Hankel operators,and invariant A∞ weights, Rev. Mat. Ibero. 13 (1997), 319-360.Dept. of Mathematics, Trinity College, Hartford, CT 06106-3100, USAE-mail address: david.cruzuribe@mail.trincoll.eduDep. de Matema´ticas, Universidad Auto´noma de Madrid, 28049 Madrid, SpainE-mail address: carlos.perez@uam.es

Two-weight, weak-type norm inequalities for singular integral operators

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Two-weight, weak-type norm inequalities for singular integral operators

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