Weak type estimates for singular integrals related to a dual problem of Muckenhoupt-Wheeden

WEAK TYPE ESTIMATES FOR SINGULAR INTEGRALS RELATED TO A DUAL PROBLEM OF MUCKENHOUPT-WHEEDEN ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS P´ EREZ Abstract. A well known open problem of Muckenhoupt-Wheeden says that any Calder´on-Zygmund singular integral operator Tis of weak type (1,1) with respect to a couple of weights (w, Mw). In this paper we consider a somewhat “dual” problem: sup λ>0 λw x∈Rn:|T f(x)| Mw > λ≤cZRn |f|dx. We prove a weaker version of this inequality with M3winstead of Mw. Also we study a related question about the behavior of the constant in terms of the A1characteristic of w. 1. Introduction In 1971, C. Fefferman and E.M. Stein [8] established the following extension of the classical weak-type (1,1) property of the Hardy- Littlewood maximal operator M: (1.1) sup λ>0 λ w{x∈Rn:Mf(x)> λ} ≤ cZRn |f|Mwdx, where a weight wis supposed to be a non-negative locally integrable function and w(E) = REw(x)dx. Assume now that Tis a Calder´on-Zygmund singular integral operator. It was conjectured by B. Muckenhoupt and R. Wheeden [13] many years ago that the full analogue of (1.1) holds for T, namely, (1.2) sup λ>0 λw{x∈Rn:|Tf(x)|> λ} ≤ cZRn |f|Mwdx. This problem is open even for the Hilbert transform. In this direction, the following result can be found in [16]: 2000 Mathematics Subject Classification. 42B20, 42B25. Key words and phrases. Calder´on-Zygmund operators, Weights. The first is supported by the Spanish Ministery of Education under the program “Programa Ram´on y Cajal, 2006. The second author is supported by a fellowship from the same institution. All the authors also supported by the same institution with research grant MTM2006-05622. 1 2 ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS P´ EREZ Theorem 1.1. There is a constant c=c(n, T)such that for any weight wand for all f, sup λ>0 λw{x∈Rn:|Tf(x)|> λ} ≤ cZRn |f|M2wdx. Here Mkdenotes the operator Miterated ktimes. In fact, it is shown in [16] that M2can be replaced by the (pointwise) smaller operator M1+εwfor any ε > 0 (see Remark 4.1 below for the definition of Mαw, α≥1). We claim that (1.2) has a somehow “dual” version, namely, (1.3) sup λ>0 λw x∈Rn:|Tf(x)| Mw > λ≤cZRn |f(x)|dx. Indeed, if conjecture (1.2) holds, say for the Hilbert transform H, then by the extrapolation theorem from [6] we can derive the following inequality for any 1 < p < ∞: ZR |Hf(x)|pwdx ≤cZR |f(x)|pMw wp wdx. Then by duality we have that for any 1 < p < ∞, ZR|Hf(x)| Mw p0 wdx ≤cZR|f(x)| wp0 wdx, and hence (1.3) can be viewed as a limiting weak-type (1,1) case of the latter inequality. Estimates of the sort (1.3) are called sometimes in the literature mixed weak type. They appeared for the first time in the work of B. Muckenhoupt and R. Wheeden [14] and later on in Sawyer’s work [17]. More recently, extensions of these results can be found in [5] and [15]. We do not know how to prove (1.3) even for M2winstead of Mw. However we prove the following result. Theorem 1.2. There is a constant c=c(n, T)such that for any weight wand for all f∈L1(Rn), sup λ>0 λw x∈Rn:|Tf(x)| M3w> λ≤cZRn |f|dx. It is interesting to observe that even a weak variant of (1.2) is not known: sup λ>0 λ w{x∈Rn:|Tf(x)|> λ} ≤ ckwkA1ZRn |f|wdx. DUAL MUCKENHOUPT-WHEEDEN PROBLEM 3 We recall that wis an A1weight if there is a finite constant csuch that Mw ≤c w a.e., and where kwkA1denotes the smallest of these c. In a recent paper [11] we proved the following related result: Theorem 1.3. Let ϕ(t) = t(1 + log+t)(1 + log+log+t). There is a constant c=c(n, T)such that for any A1weight wand for all f∈ L1 w(Rn), sup λ>0 λw{x∈Rn:|Tf(x)|> λ} ≤ c ϕ(kwkA1)ZRn |f|wdx. Analogously, we do not know whether a weak variant of (1.3) is true: (1.4) sup λ>0 λ w x∈Rn:|Tf(x)| w> λ≤ckwkA1ZRn |f|dx. Theorems 1.1 and 1.2 show that in “dual” direction we have a worst result in terms of Mk. Therefore, it is natural to expect that the bound for the left-hand side of (1.4) in terms of kwkA1must be at least not better than the one in Theorem 1.3. However, we prove the following surprising result. Theorem 1.4. There is a constant c=c(n, T)such that for any A1 weight wand for all f∈L1(Rn), sup λ>0 λw x∈Rn:|Tf(x)| w> λ≤ckwkA1(1 + log+kwkA1)ZRn |f|dx. The paper is organized as follows. Section 2 contains some preliminary information about maximal operators and singular integrals. Proofs of Theorems 1.2 and 1.4 are contained in Sections 3. Several concluding remarks are given in Section 4. 2. Preliminaries 2.1. Maximal Operator. Given a locally integrable function fon Rn, the Hardy-Littlewood maximal operator Mis defined by Mf(x) = sup Q3x 1 |Q|ZQ |f(y)|dy, where the supremum is taken over all cubes Qcontaining the point x. It is well-known (see, e.g., [10, p. 175]) that (2.1) Mkf(x)sup Q3x 1 |Q|ZQ |f|log |f| |f|Q +ek−1dy, where fQ=1 |Q|RQf. From this, by H¨older’s inequality we obtain Mkf≤c(Mf)1−1 k(Mk+1f)1 k. 4 ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS P´ EREZ In particular, for k= 2 we have the following estimate (2.2) Mf M3f1 2 ≤cMf M2f. We say that a weight wsatisfies the Apcondition if there exists a constant c > 0 such that for any cube Q, ZQ wZQ w−1/(p−1)p−1≤c|Q|p. The smallest possible chere is denoted by kwkAp. Set A∞=∪p≥1Ap. We recall that Muckenhoupt’s theorem [12] says that the maximal operator Mis bounded on Lp w,1< p < ∞,if and only if w∈Ap. We mention several well-known facts about Apweights. First, it follows from definitions and from H¨older’s inequality that if w1and w2 are A1weights, then w1w1−p 2∈Ap, and (2.3) kw1w1−p 2kAp≤ kw1kA1kw2kp−1 A1. Next, if 0 < α < 1, then (Mf)α∈A1(see [4]), and (2.4) k(Mf)αkA1≤cn,α. Let Mc wbe the weighted centered maximal operator defined by Mc wf(x) = sup Q3x 1 w(Q)ZQ |f(y)|w(y)dy, where the supremum is taken over all cubes Qcentered at x. By the Besicovitch covering theorem, (2.5) w{x∈Rn:Mc wf(x)> λ} ≤ cn λkfkL1 w. It is easy to see that for any x∈Rnone has Mf(x)≤cnMc w(f/w)(x)Mw(x). This along with (2.5) implies the following. Proposition 2.1. For any weight wand for any f∈L1(Rn),    Mf Mw  L1,∞ w ≤ckfkL1. DUAL MUCKENHOUPT-WHEEDEN PROBLEM 5 2.2. Calder´on-Zygmund operators. Let K(x, y) be a locally integrable function defined off the diagonal x=yin Rn×Rn, which satisfies the size estimate (2.6) |K(x, y)| ≤ c |x−y|n and, for some ε > 0, the regularity condition (2.7) |K(x, y)−K(z, y)|+|K(y, x)−K(y, z)| ≤ c|x−z|ε |x−y|n+ε, whenever 2|x−z|<|x−y|. A linear operator T:C∞ 0(Rn)→L1 loc(Rn) is a Calder´on-Zygmund operator if it extends to a bounded operator on L2(Rn), and there is a kernel Ksatisfying (2.6) and (2.7) such that Tf(x) = ZRn K(x, y)f(y)dy for any f∈C∞ 0(Rn) and x6∈ supp(f). We shall need the following estimate due to R. Coifman [2, 3]: for any 0 < p < ∞and for w∈A∞, (2.8) ZRn |Tf|pwdx ≤cZRn (Mf)pwdx. The following theorem has been recently proved in [11]. Theorem 2.2. Let 1<p<∞and let νp=p2 p−1log e+1 p−1. There is a constant c=c(n, T)such that for any A1weight w, (2.9) kTkLp(w)≤c νpkwkA1. Let T∗be the adjoint operator of T. Then T∗is also a Calder´on- Zygmund operator. Applying (2.9) to T∗instead of Tand using duality we have that (2.9) is equivalent to (2.10) kTkLp0(w1−p0)≤c νpkwkA1, where, as usual, 1/p + 1/p0= 1. 3. Proofs of main results Proof of Theorem 1.2. We start with some ideas used in [5]. Fix p > 1. Let Sf =Mf(Mw)1−1/2p (Mw)1−1/2p. Note that by (2.4) along with (2.3), a weight (Mw)1−r(1−1 2p)belongs to Arwith corresponding constants independent of w. Hence, by the 6 ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS P´ EREZ Muckenhoupt theorem [12], Sis bounded on Lr Mw for any r > 1. Therefore, by the Marcinkiewicz interpolation theorem [1, p. 225], Sis bounded on Lp0,1 Mw. We now apply the Rubio de Francia algorithm (see [9]) to define the operator Rby Rh(x) = ∞ X j=0 Sjh(x) (2K)j, where Kis the “norm” of Son Lp0,1 Mw. It is easy to see that (a)h(x)≤ Rh(x); (b)kRhkLp0,1(Mw)≤2khkLp0,1(Mw); (c)S(Rh)(x)≤2KRh(x). It follows from the last property that (Rh)(Mw)1−1/2p∈A1with the A1constant bounded by 2K. Using this fact and since Mw M3w≤1, we conclude by (2.3) that (3.1) (Rh)Mw (M3w)1 p ≤(Rh)(Mw)1−1 2p (M3w)1 2p ∈A2 with the A2constant depending only on pand n. Observe now that for any p > 1 we have     |Tf| M3w   L1,∞ w ≤    |Tf| M3w   L1,∞ Mw =    |Tf| M3w1/p     p Lp,∞ Mw . Next, by duality,     |Tf| M3w1/p    Lp,∞ Mw = sup ZRn |Tf|1 phMw (M3w)1 p dx, where the supremum is taken over all non-negative h∈Lp0,1 Mw with khkLp0,1 Mw = 1.Applying (3.1) and (2.8), we obtain ZRn |Tf|1 phMw (M3w)1 p dx ≤ZRn |Tf|1 p(Rh)Mw (M3w)1 p dx ≤cZRn Mf(x)1 p(Rh)(Mw)1−1 2p (M3w)1 2p dx. DUAL MUCKENHOUPT-WHEEDEN PROBLEM 7 From the last estimate and from (2.2) we get ZRn |Tf|1 phMw (M3w)1 p dx ≤c  Mf M2w1/p   Lp,∞(Mw)kRhkLp0,1(Mw) ≤2c  Mf M2w1/p   Lp,∞(Mw) = 2c   Mf M2w   1/p L1,∞(Mw) (we have used also property (b) of Rhand that khkLp0,1 Mw = 1). Applying Proposition 2.1 to the last inequality completes the proof.  Proof of Theorem 1.4. We follow here the classical method of Calder´on- Zygmund although with some modifications. Fix λ > 0, and set Ωλ={x∈Rn:Mc w(f/w)(x)> λ}. Let ∪jQjbe the Whitney covering of Ωλ. Set b(x) = supj(f−fQj)χQj(x) and g(x) = f(x)−b(x). By (2.5), w(Ωλ)≤cn λkfkL1. Hence, we have to estimate wx6∈ Ωλ:|Tf| w> λ≤wx6∈ Ωλ:|Tb| w> λ/2 +wx6∈ Ωλ:|Tg| w> λ/2≡I1+I2. By [7, Ineq. (5.4), p. 92] or [9, Lemma 5, p. 413], we get I1≤2 λZRn\Ωλ |Tb(x)|dx ≤c λX jZQj |f−fQj|dx ≤c λkfkL1, where c=c(T, n). Applying (2.10) for 1 < p0<2 yields I2≤cpp0kwkp0 A1 λp0ZRn |g|p0w1−p0dx ≤cpp0kwkp0 A1 λp0 ZRn\Ωλ |f|p0w1−p0dx +X j (|f|Qj)p0ZQj w1−p0dx!. 8 ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS P´ EREZ We have that |f| ≤ cw a.e. on Rn\Ωλ, and hence ZRn\Ωλ |f|p0w1−p0dx ≤λp0−1kfkL1. Next, by properties of the Whitney covering, it is easy to see that for any cube Qjthere exists a cube Q∗ jsuch that Qj⊂Q∗ j,|Q∗ j| ≤ cn|Qj|, and the center of Q∗ jlies outside of Ωλ. Therefore, (|f|Qj)p0−1ZQj w1−p0dx ≤ kwkp0−1 A1(|f|Qj)p0−1ZQj (Mw)1−p0dx ≤ckwkp0−1 A1|Qj| |f|Q∗ j wQ∗ j!p0−1 ≤c(λkwkA1)p0−1|Qj|, which gives X j (|f|Qj)p0ZQj w1−p0dx ≤c(λkwkA1)p0−1X j |f|Qj|Qj| ≤c(λkwkA1)p0−1kfkL1. Combining the previous estimates, we obtain I2≤cpp0kwk2p0−1 A1 λkfkL1. Choose now p0= 1 + 1 log(e+kwkA1). Then we get I2≤ckwkA1log(e+kwkA1) λkfkL1. This, along with estimates for I1and for w(Ωλ), completes the proof.  4. Concluding remarks Remark 4.1.Following (2.1), we extend the definition of Mkto fractional order, setting for any α≥1, Mαf(x) = sup Q3x 1 |Q|ZQ |f|log |f| |f|Q +eα−1dy. It was shown in [16] that M2win Theorem 1.1 can be replaced by M1+εwfor any ε > 0 (with corresponding constant depending on ε). Similarly, one can show that M3win Theorem 1.2 can be replaced by M2+εwfor any ε > 0. DUAL MUCKENHOUPT-WHEEDEN PROBLEM 9 Remark 4.2.It is easy to see that actually in Theorem 1.2 we proved a stronger inequality, namely, sup λ>0 λMw x∈Rn:|Tf(x)| M3w> λ≤cZRn |f|dx. This yields an additional indication that Theorem 1.2 should be true with M2winstead of M3w. Remark 4.3.Combining ideas used in the proving Theorem 1.4 with some estimates obtained in [16], one can show that Theorem 1.2 can be improved by replacing M3wby a smaller weight (M3w)1−εwεfor any 0<ε<1/2. We emphasize, however, that a more principal question of interest if M3wcan be replaced by M2wor simply by Mw. References [1] C. 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