Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function

Sharp estimates for commutators of singular integrals via iterations of the Hardy–Littlewood maximal function Carlos P´erez J. Fourier Analysis and Applications, 3(1997), 108–146. Departmento de Matem´aticas Universidad Aut´onoma de Madrid 28049 Madrid, Spain e–mail: cp[email protected] work partially supported by DGICYT grant PB940192, Spain 1 1 Introduction and description of the main results The purpose of this paper is to obtain some sharp non standard weighted inequalities for linear and nonlinear commutators of singular integral operators. These estimates provide a further insight into the structure of these operators and in particular they reflect a higher degree of singularity as compared with the standard Calder´on– Zygmund singular integral operators. Let Tdenote a Calder´on–Zygmund singular integral operator and let Mbe the Hardy–Littlewood maximal function. According to a result of R. Coifman [C], T and Msatisfy the following a priori estimate: Let 0 < p < ∞and suppose that w∈A∞(Rn). Then the inequality ZRn|Tf(x)|pw(x)dx ≤C[w]p A∞ZRnMf(x)pw(x)dx, (1) holds for every function ffor which the left hand side is finite. This estimate plays a major role in the modern theory of weighted norm inequalities since as it is well known it follows that Tis a bounded operator on Lp(w) whenever w∈Apand p > 1. This extends the previous result of R. A. Hunt, B. Muckenhoupt and R. L. Wheeden in [HMW] whose method works only for the Hilbert Transform. Furthermore, (1) makes explicit the well known Calder´on–Zygmund principle which establishes that a singular integral operator is controlled by an appropiate maximal function. There is another aspect of Coifman’s estimate that we shall be exploiting along this paper. It concerns the two weighted inequality problem for singular integrals, say the Hilbert transform, which is completely open. Combining (1) with certain sharp two weighted inequalities for Mwe can derive two weighted estimates for T with no a priori assumption on the weight w. As a sample we quote the following inequality from [Wil] [P3]: Let Tbe a Calder´on–Zygmund singular integral operator and let 1 < p < ∞. Then, there exists a constant Csuch that ZRn|Tf(x)|pw(x)dx ≤CZRn|f(x)|pM[p]+1w(x)dx, (2) where Cis independent of wand f. 2 1.1 Higher order commutators In this paper we are going to investigate generalizations of above inequalities (1) and (2) for a large family of singular integral operators. First we shall consider the higher order commutators introduced by R. Coifman, R. Rochberg and G. Weiss in [CRW]. These are linear operators defined for appropiate functions band fand for k= 0,1,2,· · · by Tk bf(x) = Z(b(x)−b(y))kK(x, y)f(y)dy which must be understood in the usual sense. When k= 1 the operator T1 bis usually denoted by [Mb, T] = Mb◦T−T◦Mbwhere Mbis the operator defined by Mbf=b f, and bis usually called the “symbol” of the operator. These commutators have proved to be of interest in many contexts and in particular in the theory of P.D.E. We shall only mention the recent results in the theory of non divergence elliptic equations with discontinuous coefficients [CFL1] [CFL2] [DiR]. The main result from [CRW] is the following: Let 1 < p < ∞and let b∈BMO, then there exists a constant Csuch that   Tk bf  Lp(Rn)≤Ckbkk BMO kfkLp(Rn).(3) Throughout the paper Mk=M◦(k) . . . ◦Mwill denote the Hardy–Littlewood maximal operator Miterated ktimes. Following the Calder´on–Zygmund principle we shall show that the maximal operator which controls the higher order commutators Tk bfwhen bis a BMO function is Mk+1, namely in some sense we have that Tk b≈M◦(k+1) . . . ◦M when b∈BMO. This can be made precise with the following generalization of (1). 3 THEOREM 1.1 Let 0< p < ∞and let w∈A∞and b∈BMO. Then, there exists a constant Csuch that ZRn|Tk bf(x)|pw(x)dx ≤Ckbkkp BMO [w](k+1)p A∞ZRnMk+1f(x)pw(x)dx. (4) This inequality contains the well known fact that the higher order commutators are bounded on Lp(w), w∈Ap, by applying k+ 1 times Muckenhoupt’s Theorem. As we said before (4) can be used as well to get a generalization of inequality (2). THEOREM 1.2 Let 1< p < ∞and let b∈BMO. Then, there exists a constant Csuch that for each weight w ZRn|Tk bf(x)|pw(x)dx ≤Ckbkkp BMO ZRn|f(x)|pM[(k+1)p]+1w(x)dx. (5) We remark that the number of iterations of the maximal function needed in both Theorems are optimal (see §5). In fact it follows from the proof of (5) that there is a sharper estimate: ZRn|Tk bf(x)|pw(x)dx ≤Ckbkkp BMO ZRn|f(x)|pML(log L)(k+1)p−1+(w)(x)dx where  > 0, being the result false for = 0. See §2 for the definition of ML(logL)α. Observe that both estimates (4) and (5) show that the operator Tk bbecomes more singular with ksince the maximal function on the right hand side of the inequalities needs more “iterations” to balance the inequalities. Also observe that we cannot get the sharp case (5) iterating from the case k= 1. Before continuing, let us point out that M. Wilson [Wil] was the first author who derived an estimate such as (5) for singular integrals of convolution type T0 b=Tbut only on the range 1 < p ≤2. However, Wilson’s approach is interesting because is direct and based on sharp weighted estimates for smooth Littlewood–Paley square functions using as a key step a deep result by T. Wolff [CWW] concerning the behavior of the square functions on L∞. Our method is by duality having the advantages that first covers the full range 1< p < ∞and second it is flexible enough to be applied to a wider class of operators such as Tk brather than T. Let us give an outline of the proof of Theorem 1.2 which is based on the following steps and which seems to be general enough to be applicable to other (linear) operators: 4 1. For simplicity denote Tk bby Tand [(k+ 1)p] + 1 by k(p). Now, instead of proving directly (5) we consider the corresponding (equivalent) dual inequality, namely ZRn|Tf(x)|p0(Mk(p)w(x))1−p0dx ≤CZRn|f(x)|p0 w(x)1−p0dx (6) since the adjoint operator to Tk bis essentially the same. 2. After observing that (Mk(p)w)1−p0∈A∞(in fact it belongs to RH∞) we apply the Calder´on–Zygmund principle: we replace the singular integral by a maximal type operator, namely Mk+1 in our case using Theorem 1.1: ZRn|Tf(x)|p0(Mk(p)w(x))1−p0dx ≤CZRnMk+1f(x)p0(Mk(p)w(x))1−p0dx. (7) 3. Therefore everything is reduced to showing a sharp two weighted norm inequalities for the maximal operator Mk+1 ZRnMk+1f(x)p0(Mk(p)w(x))1−p0dx ≤CZRn|f(x)|p0 w(x)1−p0dx. (8) 1.2 The Nonlinear commutator The second commutator that we are going to consider was introduced by R. Rochberg and G. Weiss in [RW]. This nonlinear operator is defined for appropriate functions by f→Nf =T(flog |f|)−Tf log |Tf|. Nis homogeneous and can be written as a commutator [Ω, T ] = T◦Ω−Ω◦Twhere Ω denotes the operation Ωf=flog |f|. There is a growing interest in studying this operator due to its relationship with the Jacobian mapping and with nonlinear P.D.E. as shown in [IS] [GI] (see also [M]). The main result from [RW] is the following: Let 1 < p < ∞, then there exists a constant Csuch that kNfkLp(Rn)≤CkfkLp(Rn).(9) 5 The theory developed in [RW] is very general. It shows, for instance, that the singular integral Tmay be replaced by any linear operator bounded on Lpi(Rn), i= 1,2, with 1 < p1<p<p2<∞. However, to derive Aptype estimates for N such a general framework does not seem to be suitable. We shall be using a different approach based on real variable techniques and in particular on the theory of Ap weights combined with some of the estimates obtained above for the linear commutator [Mb, T]. Furthermore and trying to follow the Calder´on–Zygmund principle again, we show that the maximal operator which controls Nis the Hardy–Littlewood maximal function iterated twice, namely N≈M◦M, expression which more precisely means the following: THEOREM 1.3 Suppose that 0<p<∞and that w∈A∞. Then, there exists a constant Csuch that ZRn|Nf(x)|pw(x)dx ≤C[w]p A∞ZRnM2f(x)pw(x)dx, (10) As an immediate consequence we have the following corollary. COROLLARY 1.4 Let 1< p < ∞and let w∈Ap. Then, there exists a constant Csuch that ZRn|Nf(x)|pw(x)dx ≤C[w]3p ApZRn|f(x)|pw(x)dx, (11) Contrary to what we did for the linear commutator Tk bwe cannot apply Theorem 1.3 to derive for Na result in the spirit of Theorem 1.2. The method sketched above breakdowns due to the nonlinearity of N. However and by a direct approach we can still deduce a corresponding estimate. THEOREM 1.5 Suppose that 1< p < ∞. Then, there exists a constant Csuch that for each weight w ZRn|Nf(x)|pw(x)dx ≤CZRn|f(x)|pM[2p]+1w(x)dx. (12) 6 To get this estimate whe show that there is a relationship between Nand the linear commutator [Mb, T] and consequently with M◦M=M2. The observation is that Ncan be written using the linearity of Tas follows (see §4): Nf =T(flog |f| Mf )+[Mlog Mf , T](f)−Tf log |Tf| Mf =N1f+N2f+N3f. Observe that the symbol of the operator N2is the operation b=b(f) = log Mf which is a BMO function with a constant independent of f. 2 Some preliminaries and notation We shall introduce in this section some of the necessary tools that we need to prove our results. Recall that a function B: [0,∞)→[0,∞) is called a Young function if it is continuous, convex and increasing satisfying B(0) = 0 and B(t)→ ∞ as t→ ∞. We define the B–average of a function fover a cube Qby means of the Luxemburg norm kfkB,Q = inf{λ > 0 : 1 |Q|ZQ B |f(y)| λ!dy ≤1},(13) and recall the following generalization of H¨older’s inequality: 1 |Q|ZQ|f(y)g(y)|dy ≤ kfkB,Q kgk¯ B,Q ,(14) where ¯ Bis the complementary Young function associated to B. There is a further generalization which turns out to be useful for our purposes (see [O1]): Let A,B, Cbe Young functions such that A−1(t)·B−1(t)≤C−1(t), then kfgkC,Q ≤2kfkA,Q kfkB,Q (15) We define a natural maximal operator associated to the Young function associated to B. 7 DEFINITION 2.1 For each locally integrable function fthe maximal operator MBis defined by MBf(x) = sup x∈Q kfkB,Q , where the supremum is taken over all the cubes containing x. The main examples that we are going to be using are B(t) = t(1 + log+t)α, α > 0, with maximal function denoted by ML(logL)α. The complementary Young function is given by ¯ B(t)≈et1/α with corresponding maximal function denoted by Mexp(L1/α). The boundedness properties of MBwill play a central role to derive sharp two weighted estimates. We need the following class of Young functions. DEFINITION 2.2 Let 1< p < ∞. We say that a doubling Young function B satisfies the Bpcondition if there is a positive constant csuch that Z∞ c B(t) tp dt t≈Z∞ c tp0 ¯ B(t)!p−1dt t<∞. This condition provides with a characterization of those maximal operators MB which are bounded on Lp(Rn), 1 < p < ∞. In fact, we have the following Theorem whose proof can be found in [P1]. THEOREM 2.3 Let 1<p<∞. Suppose that Bis a doubling Young function. Then the following are equivalent. i) B∈Bp; (16) ii) there is a constant csuch that ZRnMBf(x)pdx ≤cZRn|f(x)|pdx (17) for all functions f; iii) there is a constant csuch that ZRnMBf(x)pw(x)dx ≤cZRn|f(x)|pMw(x)dx (18) 8 for all functions fand all weights w; iv) there is a constant csuch that ZRnMf(x)pw(x) [M¯ B(u1/p)(x)]pdx ≤cZRn|f(x)|pMw(x) u(x)dx, (19) for all functions fand all weights wand u. In the proof of Theorem 1.2 and for p > 1 we shall be working with Young functions of the form B(t)≈tp(log t)−1−which satisfy the Bpcondition and therefore the associated maximal operators MLp(log L)−1−are bounded on Lp(Rn). We conclude this section with a corollary of this Theorem that will be used later on. The result can be seen as a weighted inequality “dual” to the classical Fefferman–Stein inequality ZRnMf(x)pw(x)dx ≤cZRn|f(x)|pMw(x)dx. If Mwere a linear operator this inequality would imply ZRnMf(x)p0Mw(x)1−p0dy ≤cZRn|f(x)|p0 w(x)1−p0dx, which is false in general, however we have the following sharp replacement. COROLLARY 2.4 Let 1<p<∞and let w, u be weights. Then there exists a constant Cindependent of the weights such that ZRnMf(x)p0u(x) M[p]+1w(x)p0−1dx ≤cZRn|f(x)|p0Mu(x) w(x)p0−1dx (20) for all f. In particular if u∈A1 ZRnMf(x)p0u(x) M[p]+1w(x)p0−1dx ≤c[u]A1ZRn|f(x)|p0u(x) w(x)p0−1dx (21) Proof: By part iv) of the Theorem we have that B∈Bp0if and only if ZRnMf(x)p0w(x) [M¯ B(u(p0−1)/p0)(x)]p0dx ≤cZRn|f(x)|p0Mw(x) u(x)p0−1dx, 9 ≤C[w]p A∞ZRn(M2f)pw. For the last term we split Rnin two disjoint sets Aand Bwhere A={y∈Rn:|Tf(y)| ≤ Mf(y)}and B={y∈Rn:|Tf(y)|> Mf(y)}. Writing N3f=Mf |Tf| Mf log |Tf| Mf = Using in Bthat log t≤Ct ,t > 1,  > 0 we have the following ZRn|N3f|pw≤CZA(Mf)pw+CZB|Tf|p(+1) (Mf)− p w We would like to apply again Theorem 1.1 with k= 0. To do this we must show that w(Mf)− p w∈A∞for small enough and with a constant independent of f. (recall that is still available). Indeed, since w∈A∞w∈Aqfor some q > 1 and by the factorization (cf. [GCRdF] p. 436) theorem w=w1w1−q 2where w1and w2 are A1weights. Then w(Mf)− p =w1w1−q 2(Mf)− p =w1(w2(Mf) p q−1)1−q. By the factorization theorem it is enough to show that w2(Mf) p q−1∈A1for small enough. To do this we fix a cube Q, and arbitrary a.e. x∈Q. Then 1 |Q|ZQ w2(Mf) p q−1≤(1 |Q|ZQ wr 2)1/r (1 |Q|ZQ(Mf) r0p q−1)1/r0.(27) Now, since w2∈A∞we can pick r > 1 such that we can continue with C |Q|ZQ w2(1 |Q|ZQ(Mf) r0p q−1)1/r0, and if we pick with 0 <  < q−1 p r0then (Mf) r0p q−1∈A1and then this is less or equal than C |Q|ZQ w2(Mf(x))  p q−1≤C w2(x) (Mf(x))  p q−1. An important observation is that the A∞norm does not depend on f.2 Proof of Theorem 1.5 Proceeding as before we have 16 kNfkLp(w)≤ kN1fkLp(w)+kN2fkLp(w)+kN3fkLp(w), and the proof of the two first pieces are similar to the previous case. Recall that there is no assumption on w. For N1we combine Theorem 1.2 for k= 0, the fact that |tlog t| ≤ 1 e, 0 < t ≤1, together with the classical Fefferman–Stein inequality ZRn(Mf)pw≤cZRn|f|pMw to obtain ZRn|N1f|pw=ZRn|T(Mf f Mf log(|f| Mf )| p w ≤CZRn(Mf)pM[p]+1w≤CZRn|f(y)|pM[p]+2w ≤CZRn|f|pM[2p]+1w. For N2we use Theorem 1.2 since de BMO norm of log Mf is independent of f ZRn|N2f|pw=ZRn|[log Mf, T]f|pw ≤ klog Mfk2p BMO ZRn|f|pM[2p]+1w≤CZRn|f|pM[2p]+1w. For the last term N3we start as above with ZRn|N3f|pw≤CZRn(Mf)pw+CZRn|Tf|p(+1) (Mf)− p w, ≤CZRn|f|pMw +CZRn|Tf|p(+1) (Mf)− p w. The key point of the proof is to understand the last term. We are tempted in applying Theorem 1.1 with k= 0 replacing Tby Mwhich would finish the proof of the Theorem; however, there is no assumption on win such a way that we cannot say that the weight on the right hand side, namely (Mf)− p w, is an A∞weight. We may argue as follows. Let p=p(+ 1), then there exist a function g∈L(p)0with unit norm such that ZRn|Tf|p(Mf)− p w1 p=  Tf (Mf)− +1 w1 p  Lp(Rn)=ZRnTf (Mf)− +1 w1 pg. 17 Since the adjoint operator T∗is also a Calder´on– Zygmund operator bounded on all the Lpspaces as well we can equal last expression to ZRnf T∗(g(Mf)− +1 w1 p) = ZRnf(Mkw)1 p (Mf) +1 T∗(g(Mf)− +1 w1 p)(Mf) +1 (Mkw)1 p ≤ ZRn|f|pMkw (Mf) +1 p!1 p ZRn|T∗(g(Mf)− +1 w1 p)|(p)0(Mf) +1 (p)0 (Mkw)(p)0 p  1 (p)0 =I×II, where kis an integer to be chosen in a moment. To estimate Iwe simply use the Lebesgue differentiation Theorem I= ZRn|f|p  |f|pMkw (Mf)p  !1 p ≤ZRn|f|pMkw1 p, where kis still available. For the last term II we are going to replace T∗by the Hardy–Littlewood maximal function using again Theorem 1.1 with k= 0 and since T∗is also a Calder´on–Zygmund operator. All we have to do is to show that the weight u=(Mf) +1 (p)0 (Mkw)(p)0 p = (Mf) +1 (p)0(Mkw)1−(p)0 is an A∞weight with constant independent of f. To do this observe first that (Mf) +1 (p)0∈A1since  +1 (p)0= +1/p0<1 and (Mkw)1−(p)0∈RH∞by Lemma 4.2 where the constants are independent of both fand w. Therefore u∈A∞by Lemma 4.1 above and we have applying Theorem 1.1 that II ≤C ZRnM(g Mf− +1 w1 p)(p)0(Mf) +1 (p)0 (Mkw)(p)0−1! 1 (p)0 . Finally we can apply Corollary 2.4 with k= [p] + 1 using as we pointed out above that (Mf) +1 (p)0∈A1. Then II ≤C ZRn|g|(p)0(Mf)− +1 (p)0w(p)0 p(Mf) +1 (p)0 w(p)0−1! 1 (p)0 18 =CZRn|g|(p)01 (p)0 =C. Combining all these inequalities we get that ZRn|Tf|p(+1) Mf− p w≤CZRn|f|pM[p]+1w=CZRn|f|pM[p]+1w with small enough. Therefore we have ZRn|N3f(y)|pw(y)dy ≤ZRn|f(y)|pM[p]+1w(y)dy which combined with the estimates for N1and N2yield the final result. Observe that the piece corresponding to N3behaves more as the usual singular integral and that the “worst” piece corresponds to N2.2 5 A counterexample We end the paper by showing that Theorem 1.2, and consequently the others, is optimal. Consider the classical Hilbert transform Hf(x) = pv ZR f(y) x−ydy, and let m= 1,2,· · · be the largest exponent for which the following inequality does not hold ZRn|Hk bf(x)|pw(x)dx ≤Ckbkkp BMO ZRn|f(x)|pMmw(x)dx. (28) By duality this is equivalent to showing ZRn|Hk bf(x)|p0 Mmw(x)1−p0dx ≤Ckbkkp0 BMO ZRn|f(x)|p0 w(x)1−p0dx Consider the BMO function b(x) = log |x|and let f=w=χ(0,1) so that the right hand is equal to a finite constant. 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