Sharp weighted endpoint estimates for commutators of singular integral operators

Michigan Math. J. 49 (2001) Sharp Weighted Endpoint Estimates for Commutators of Singular Integrals Carlos Pérez & Gladis Pradolini 1. Introduction and Statements of the Main Result The main purpose of this paper is to improve the main result in [P2] by means of a direct proof that avoids the classical good-λtechnique considered there. The good-λmethod, introduced by Burkholder and Gundy in [BG], is a powerful tool buthasthedisadvantagethatitisessentiallyadaptedtomeasuressatisfyingtheA∞ condition, such as the Lebesgue measure. The approach we consider here is more related to the classical argument of Calderón and Zygmund for proving that singular integral operators satisfy the weak-type (1,1)-property, an approach whose advantageis that itallows usto considermore general measure. The method, however, must be different because commutators of singular integral operators with BMO functions are not of weak-type (1,1), as shown in [P2]. Let bbe a locally integrable function on Rn,usually called the symbol, and let Tbe a Calderón–Zygmund singular integral operator (see [C] or [J]). Consider the commutator operator [b, T ] defined for, say, smooth functions fby [b, T ]f=bT (f ) −T(bf). (1) A now classical result of Coifman, Rochberg, and Weiss [CRW] states that [b, T ] is a bounded operator on Lp(Rn), 1<p<∞,when bis a BMO function. In fact, BMO is also a necessary condition for the commutator [b, R] to be bounded on Lp(Rn), where R=(R1,...,Rn)is the vector-valued Riesz transform. We will always assume that b∈BMO(Rn)unless otherwise noted. None of the different proofs of this result follows the usual scheme of the classical Calderón–Zygmund theory of singular integral operators T. Indeed, the key result in this theory is that any of these operators satisfies the weak-type (1,1)- property, which is derived from the assumption that Tis bounded on L2(Rn)combined with a mild regularity of the kernel. Once the weak-type (1,1)-inequality is obtained, interpolation and duality yield the boundedness of the operator on Lp(Rn)for all 1 <p<∞.However, simple examples show that commutators (with BMO symbols) fail to be of weak-type (1,1), as found in [P2]. To remedy Received May 3, 2000. Revision received October11, 2000. This research was carried out during a stay at the Universidad Autónoma de Madrid. Research of the first author was partially supported by DGESIC grant PB98-0106, Spain. Research of the second author was supported by Universidad Nacional del Litoral (FOMEC) and CONICET, República Argentina. 23 24 Carlos Pérez & Gladis Pradolini the situation, it is shown there that commutators satisfy a “L(logL)”-type estimate. To be precise, we have the following result. Theorem 1.1 [P2]. Let Tbe any Calderón–Zygmund singular integral operator. Then there exists a positive constant Cdepending upon the BMO norm of b such that, for all functions fand all λ>0, |{y∈Rn:|[b, T ]f(y)|>λ}| ≤ CZRn |f(y)| λ1+log+|f(y)| λdy. (2) Theproofof this estimateisbased on showingthat there is anintimaterelationship between commutators and iterations of the Hardy–Littlewood maximal function (in this case, M2=MBM)via the good-λtechnique of Burkholder and Gundy [BG]. More precisely, if we let 8(t) =t(1+log+t) and let wbe a weight satisfying the A∞condition, then there exists a positive constant Cdepending on the BMO constant of bsuch that, for any smooth function with compact support f, sup t>0 1 8(1/t)w({y∈Rn:|[b, T ]f(y)|>t}) ≤C[w]A∞sup t>0 1 8(1/t)w({y∈Rn:M2f(y)>t}). (3) Using this estimate with w=1 and analyzing the behavior of M2,we obtain the desired estimate (2)—where, in fact, the Lebesgue measure can be replaced by any weight function satisfying the A1condition. The Lpversions of these estimates and their consequences are further exploited in [P3]. As mentioned before, we provide a different proof of (2) whose advantage is that it allows us to derive a sharp two-weight inequality that is similar in spirit to the following one for Calderón–Zygmund singular integral operators. Theorem 1.2 [P1]. Let Tbe any Calderón–Zygmund operator and let ε>0. Then, for any weight w, function f, and t>0,there is a constant Cεsuch that w({x∈Rn:|Tf (x)|>t})≤Cε tZRn |f(x)|ML(logL)ε(w)(x) dx. (4) The point here is that no assumption on the weight is assumed. Here MA=MA(L) denotes a maximal-type function defined by the expression MA(L)f(x) =sup Q3x kfkA,Q, where Ais anyYoung function and kfkA,Qdenotes the A-average over Qdefined by means of the Luxembourg norm kfkA,Q=infλ>0: 1 |Q|ZQ A|f| λdx ≤1.(5) For our applications, the main examples are given by A(t ) =t(1+log+t)α, α≥0. We will consider a more general version of (1) denoted by Tm b(m =0,1,2,...) and usually called higher-order commutators. They are defined by the formula Sharp Weighted Endpoint Estimates for Commutators of Singular Integrals 25 Tm b=[b,...,[b, T ]] |{z } (m times) ; in the particular case of Calderón–Zygmund operators, they can be expressed by means of its kernel K: Tm bf(x) =ZRn (b(x) −b(y))mK(x,y)f(y)dy, where fis an appropriate test function. As usual, we assume that the kernel K satisfies the so-called standard estimates (cf. [C] or [J]). Our result is the following. Theorem 1.3. Let Tbe a Calderón–Zygmund singular integral operator, and let b∈BMO and ε>0.Then there exists a positive constant Csuch that w({x∈Rn:|Tm bf(x)|>λ}) ≤CZRn 8mkbkm BMO |f(x)| λML(logL)m+ε(w)(x) dx, (6) where 8m(t) =t(1+log+t)m.The constant Cis independent of the weight w, the function f, and λ>0. Observethatthereisnorestrictionontheclassofweightsconsidered. Observealso that, since 8mis submultiplicative (i.e., 8m(ab) ≤C8m(a)8m(b) with a, b ≥ 0), we have w({x∈Rn:|Tm bf(x)|>λ}) ≤C8m(kbkm BMO)ZRn 8m|f(x)| λML(logL)m+ε(w)(x) dx. Inequalities similar to (6) have turned out to be very useful in the study of the two-weight problemfor singularintegraloperators (see [CP1; CP2]). On the other hand, it would be interesting to know whether or not this inequality holds when ε=0. 2. Some Preliminaries and Notation In this section we summarize a few facts about Orlicz spaces. (For more information, see Bennett and Sharpley [BS] or Rao and Ren [RR].) A function B:[0,∞)→[0,∞)isadoublingYoung function if: (a) it is continuous, convex, and increasing; (b) B(0)=0 and B(t) →∞as t→∞;and (c) it satisfies B(2t) ≤CB(t) for all t>0. For Orlicz norms we are usually concerned about the behavior ofYoung functions for tlarge. Given two functions Band C, we write B(t) ∼ =C(t) if B(t)/C(t) is bounded and bounded below for t≥c>0. Recall that we defined the localized Luxembourg norm by equation (5); an equivalent norm that is often useful in calculations is due to Krasnosel’ski˘ı and Ruticki˘ı [KR, p. 92] (also see [RR, p. 69]): 26 Carlos Pérez & Gladis Pradolini kfkA,Q≤inf µ>0µ+µ |Q|ZQ A|f| µdx≤2kfkA,Q.(7) Given aYoung function A, we use ¯ Ato denote the complementaryYoung function associated to A;it has the property that, for all t>0, t≤A−1(t) ¯ A−1(t) ≤2t. The basic property that we will use is the following generalized Hölder inequality: 1 |Q|ZQ |fg|≤2kfkA,Qkgk¯ A,Q.(8) In particular, we shall work with A(t ) =t(1+log+t)m,m=1,2,..., with maximal function denoted by ML(logL)m.The complementery Young function is given by ¯ A(t ) ≈exp(t1/m), with the corresponding maximal function denoted by Mexp L1/m . The first generalizedYoung inequality states that A−1(t) ·B−1(t) ≤C−1(t) for t>0;it follows that C(st) ≤A(s) +B(t) (9) holds for all s,t > 0. 3. Proof of the Theorem In this section we prove Theorem 1.3 by induction from the case m=1.We will use the following strong-type version of our estimate derived in [P3]. Theorem 3.1 [P3]. Let Tbe any Calderón–Zygmund singular integral operator, and let 1<p<∞and b∈BMO.Then for each δ>0there exists a positive constant C=Cδsuch that, for all functions g, ZRn |Tm bg(x)|pw(x) dx ≤Cδkbkmp BMO ZRn |g(y)|pML(logL)(m+1)p−1+δ(w)(y) dy. (10) 3.1. The Case m=1 A simple homogeneity shows that we may assume kbkBMO =1.Given that assumption, we need only show that w({x∈Rn:|[b, T ]f(x)|>λ})≤CZRn 8|f(x)| λML(logL)1+ε(w)(x) dx, where 8(t) =81(t) =t(1+log+t). We considerthe standardCalderón–Zygmund decompositionof fatlevelλand obtain a collection of dyadic non-overlapping cubes Qj=Qj(xQj,r j)that satisfy λ< 1 |Qj|ZQj |f|≤2nλ. (11) Sharp Weighted Endpoint Estimates for Commutators of Singular Integrals 27 We set =λ=SjQj;then |f(x)|≤λa.e. x∈Rn\. We write f= g+h, where gis defined by g(x) =f(x) if x∈Rn\, fQjif x∈Qj. As usual, we use the notation fQ=1 |Q|RQffor a locally integrable function f and a cube Q. Observe that |g(x)|≤2nλa.e. We split the “bad part” has h=Pjhj,where hj(x) =(f(x) −fQj)χQj(x). We will use the notation w∗(x) =w(x)χRn\˜ (x) and wj(x) =w(x)χRn\3Qj, where ˜ Qj=3Qjand ˜ =Sj˜ Qj.Then w({x∈Rn:|[b, T ]f(x)|>λ}) ≤w({x∈Rn\˜ :|[b, T ]g(x)|>λ/2})+w( ˜ ) +w({x∈Rn\˜ :|[b, T ]h(x)|>λ/2}) =I+II+III. As we will see from the proof, part I (precisely the piece associated to the “good part”) is the one that carries a higher degree of singularity. Now we use Theorem 3.1, with m=1and with p, δ such that 1 <p<1+ε/2 and δ=ε−2(p −1)> 0.Then I≤C λpZRn |[b, T ]g(x)|pw∗(x)dx ≤C λpZRn |g(x)|pML(logL)1+ε(w∗)(x) dx ≤C λZRn |g(x)|ML(logL)1+ε(w∗)(x) dx =C λZRn\ |f(x)|ML(logL)1+ε(w)(x) dx +Z |g(y)|ML(logL)1+ε(w∗)(y) dy. It is clear that we need only estimate the second term in the last expression; to do so, we use the following fact: ForarbitraryYoungfunctionA, nonnegativemeasurewwithMAw(x) < ∞a.e., cube Q, and R>1,we have MA(χRn\RQw)(y) ≈MA(χRn\RQw)(z) for each y,z ∈Q;hence MA(χRn\RQw)(y) ≈inf y∈QMA(χRn\RQw)(y) (12) for each y∈Q. This is an observation whose proof follows exactly as for the case of the Hardy– Littlewood maximal operator M, which corresponds to the case A(t ) =t(see e.g. [GR, p. 159]). 28 Carlos Pérez & Gladis Pradolini Hence we can continue estimating the second term with Z |g(x)|ML(logL)1+ε(w∗)(x) dx ≤X jZQj |fQj|ML(logL)1+ε(wj)(x) dx =X jZQj |f(x)|dx1 |Qj|ZQj ML(logL)1+ε(wj)(x) dx ≤CX jZQj |f(x)|dxinf Qj ML(logL)1+ε(wj) =CX jZQj |f(x)|ML(logL)1+ε(w)(x) dx ≤CZRn |f(x)|ML(logL)1+ε(w)(x) dx. ForIIwehave II =w( ˜ ) ≤CX j w( ˜ Qj) |˜ Qj||Qj|≤C λX j w( ˜ Qj) |˜ Qj|ZQj |f(x)|dx ≤C λX jZQj |f(x)|Mw(x) dx ≤C λZRn |f(x)|Mw(x) dx. Observe that this part is smoother than I since we obtain a smaller operator Mon the right-hand side. Similarly, part III is smoother than I but rougher than II, as we now show. Indeed, first note that [b, T ]h(x) =X j [b, T ]hj(x) =X j (b(x) −bQj)T hj(x) −X j T ((b −bQj)h j)(x), where (as before) bQ=1 |Q|RQb. Then III ≤wx∈Rn\˜ :X j (b(x) −bQj)T hj(x)>λ 4 +wx∈Rn\˜ :X j T ((b −bQj)h j)(x)>λ 4 =A+B. Sharp Weighted Endpoint Estimates for Commutators of Singular Integrals 29 Using the standard estimates of the kernel K, we have A≤C λZRn\˜ X j |b(x) −bQj||Th j(x)|w(x) dx ≤C λX jZRn\3Qj |b(x) −bQj|w(x) ZQj |hj(y)||K(x −y) −K(x −xQj)|dy dx ≤C λX jZQj |hj(y)|ZRn\3Qj |K(x −y) −K(x −xQj)||b(x) −bQj|wj(x)dx dy ≤C λX jZQj |hj(y)| ∞ X k=1 Z2krj≤|x−xQj|<2k+1rj |y−xQj| |x−xQj|n+1|b(x) −bQj|wj(x)dx dy ≤C λX jZQj |hj(y)|dy∞ X k=1 2−k (2k+1rj)nZ|x−xQj|<2k+1rj |b(x) −bQj|wj(x)dx. To control the sum on k, we use standard estimates together with the generalized Hölder inequality and the John–Nirenberg theorem. Indeed, if y∈Qjthen we have ∞ X k=1 2−k (2k+1rj)nZ|x−xQj|<2k+1rj |b(x) −bQj|wj(x)dx ≤C ∞ X k=1 2−k (2k+1rj)nZ2k+1Qj |b(x) −b2k+1Qj|wj(x)dx + ∞ X k=1 2−k (2k+1rj)nZ2k+1Qj |b2k+1Qj−bQj|wj(x)dx ≤C ∞ X k=1 2−kkb−b2k+1QjkexpL,2k+1QjkwjkLlogL,2k+1Qj + ∞ X k=1 2−k(k +1)M(wj)(y) ≤CML(logL)(wj)(y) ∞ X k=1 2−k+M(w j)(y) ∞ X k=1 2−k(k +1)  ≤CMLlogL(wj)(y). Then we can continue the estimate of Ausing (12) as follows: 30 Carlos Pérez & Gladis Pradolini A≤C λX jZQj |hj(y)|MLlogL(wj)(y) dy ≤C λX jZQj |f(y)|MLlogL(w)(y) dy +X jZQj |fQj|MLlogL(wj)(y) dy ≤C λZRn |f(y)|MLlogL(w)(y) dy +X jZQj |f(x)|dx 1 |Qj|ZQj MLlogL(wj)(y) dy ≤C λZRn |f(y)|MLlogL(w)(y) dy +X jZQj |f(x)|MLlogL(w)(x) dx ≤C λZRn |f(y)|MLlogL(w)(y) dy. To estimate B, we combine inequality (4) for singular integrals together with (again) observation (12): B=w∗x∈Rn:TX j (b −bQj)h j(x)>λ 4 ≤C λZRnX j (b(x) −bQj)h j)(x)ML(logL)ε(w∗)(x) dx ≤C λX jZQj |b(x) −bQj||f(x)−fQj|ML(logL)ε(wj)(x) dx ≤C λX j inf Qj ML(logL)ε(wj)(x)ZQj |b(x) −bQj||f(x)|dx +ZQj |b(x) −bQj||fQj|dx =B1+B2. The estimate for B2is simple since, by (12), B2=C λX j inf Qj ML(logL)ε(wj)(x) ZQj |b(x) −bQj||fQj|dx ≤C λX j 1 |Qj|ZQj |b(x) −bQj|ZQj |f(x)|ML(logL)ε(wj)(x) dx ≤CZRn |f(x)|ML(logL)ε(w)(x) dx. Sharp Weighted Endpoint Estimates for Commutators of Singular Integrals 31 For B1we have, by the generalized Hölder inequality (8), B1=C λX j inf Qj ML(logL)ε(wj)(x) ZQj |b(x) −bQj||f(x)|dx ≤C λX j inf Qj ML(logL)ε(wj)(x)|Qj|kfkLlogL,Qj. Now, combining formula (7) with (11) and recalling that 8(t) =t(1+log+t), we have 1 λ|Qj|kfkLlogL,Qj≤1 λ|Qj|inf µ>0µ+µ |Qj|ZQj 8|f(x)| µdx ≤|Qj|+ZQj 8|f(x)| λdx ≤1 λZQj |f(x)|dx +ZQj 8|f(x)| λdx ≤2ZQj 8|f(x)| λdx. Then B1≤CZQj 8|f(x)| λML(logL)ε(wj)(x) dx ≤CZRn 8|f(x)| λML(logL)ε(w)(x) dx. This concludes the proof of the case m=1. 3.2. The General Case We will use an induction argument and will omit some technical arguments that are similar to the case m=1.Again, a simply homogeneity argument using that Tm b(f/kbkm BMO)=Tm b/kbkBMO(f ) shows that we may assume kbkBMO =1.We consider again the Calderón–Zygmund decomposition of fat level λ. Then, with the same notation as in the proof of the case m=1,we have w({y∈Rn:|Tm bf(y)|>λ})≤w({y∈Rn\˜ :|Tm bg(y)|>λ/2})+w( ˜ ) +w({y∈Rn\˜ :|Tm bh(y)|>λ/2}) =I+II+III. From (10) with pand δsuch that 1<p<1+ε/(m +1)and δ=ε−(m +1)(p −1)>0,