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 PÉREZ Abstract. A ell knon open problem of Muckenhoupt-Wheeden says that any Calderón-Zygmund singular integral operator T is of eak type (1, 1) ith respect to a couple of eights (, ). In this paper e consider a somehat dual problem: sup λ x T f(x) : > λ } c f dx. We prove a eaker version of this inequality ith M 3 instead of. Also e study a related question about the behavior of the constant in terms of the characteristic of. 1. Introduction In 1971, C. Fefferman and E.M. Stein [8] established the folloing extension of the classical eak-type (1, 1) property of the Hardy- Littleood maximal operator M: (1.1) sup λ x : Mf(x) > λ} c f dx, here a eight is supposed to be a non-negative locally integrable function and (E) = (x)dx. E Assume no that T is a Calderón-Zygmund singular integral operator. It as 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 λx : T f(x) > λ} c f dx. This problem is open even for the Hilbert transform. In this direction, the folloing result can be found in [16]: 2000 Mathematics Subject Classification. 42B20, 42B25. Key ords and phrases. Calderón-Zygmund operators, Weights. The first author is supported by the Spanish Ministry of Education under the program Programa Ramón y Cajal, 2006. The second author is supported by a felloship from the same institution. All the authors also supported by the same institution ith research grant MTM2006-05622. 1

2 ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS PÉREZ Theorem 1.1. There is a constant c = c(n, T ) such that for any eight and for all f, sup λx : T f(x) > λ} c f M 2 dx. Here M k denotes the operator M iterated k times. In fact, it is shon in [16] that M 2 can be replaced by the (pointise) smaller operator M 1+ε for any ε > 0 (see Remark 4.1 belo for the definition of M α, α 1). We claim that (1.2) has a someho dual version, namely, (1.3) sup λ x : T f(x) } > λ c f(x) dx. Indeed, if conjecture (1.2) holds, say for the Hilbert transform H, then by the extrapolation theorem from [6] e can derive the folloing inequality for any 1 < p < : ( ) p Hf(x) p dx c f(x) p dx. R Then by duality e have that for any 1 < p <, ( ) p ( Hf(x) f(x) dx c R R R ) p dx, and hence (1.3) can be vieed as a limiting eak-type (1, 1) case of the latter inequality. Estimates of the sort (1.3) are called sometimes in the literature mixed eak type. They appeared for the first time in the ork of B. Muckenhoupt and R. Wheeden [14] and later on in Sayer s ork [17]. More recently, extensions of these results can be found in [5] and [15]. We do not kno ho to prove (1.3) even for M 2 instead of. Hoever e prove the folloing result. Theorem 1.2. There is a constant c = c(n, T ) such that for any eight and for all f L 1 ( ), x : sup λ T f(x) M 3 } > λ c f dx. It is interesting to observe that even a eak variant of (1.2) is not knon: sup λ x : T f(x) > λ} c A1 f dx.

DUAL MUCKENHOUPT-WHEEDEN PROBLEM 3 We recall that is an eight if there is a finite constant c such that c a.e., and here A1 denotes the smallest of these c. In a recent paper [11] e proved the folloing 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 eight and for all f L 1 ( ), sup λx : T f(x) > λ} c ϕ( A1 ) f dx. Analogously, e do not kno hether a eak variant of (1.3) is true: } (1.4) sup λ x T f(x) : > λ c A1 f dx. Theorems 1.1 and 1.2 sho that in dual direction e have a orst result in terms of M k. Therefore, it is natural to expect that the bound for the left-hand side of (1.4) in terms of A1 must be at least not better than the one in Theorem 1.3. Hoever, e prove the folloing surprising result. Theorem 1.4. There is a constant c = c(n, T ) such that for any eight and for all f L 1 ( ), } sup λ x T f(x) : > λ c A1 (1 + log + A1 ) f dx. The paper is organized as follos. 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 f on, the Hardy-Littleood maximal operator M is defined by 1 Mf(x) = sup f(y) dy, Q x Q here the supremum is taken over all cubes Q containing the point x. It is ell-knon (see, e.g., [10, p. 175]) that (2.1) M k 1 ( f ) k 1dy, f(x) sup f log + e Q x Q Q f Q here f Q = 1 f. From this, by Hölder s inequality e obtain Q Q M k f c(mf) 1 1 k (M k+1 f) 1 k. Q

4 ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS PÉREZ In particular, for k = 2 e have the folloing estimate (2.2) ( ) 1 Mf 2 Mf c M 3 f M 2 f. We say that a eight satisfies the A p condition if there exists a constant c > 0 such that for any cube Q, ( )( ) p 1 1/(p 1) c Q p. Q Q The smallest possible c here is denoted by Ap. Set A = p 1 A p. We recall that Muckenhoupt s theorem [12] says that the maximal operator M is bounded on L p, 1 < p <, if and only if A p. We mention several ell-knon facts about A p eights. First, it follos from definitions and from Hölder s inequality that if 1 and 2 are eights, then 1 1 p 2 A p, and (2.3) 1 1 p 2 Ap 1 A1 2 p 1. Next, if 0 < α < 1, then (Mf) α (see [4]), and (2.4) (Mf) α A1 c n,α. Let c be the eighted centered maximal operator defined by f(x) c 1 = sup f(y) (y)dy, Q x (Q) here the supremum is taken over all cubes Q centered at x. By the Besicovitch covering theorem, (2.5) x : M c f(x) > λ} c n λ f L 1. It is easy to see that for any x one has Mf(x) c n M c (f/)(x)(x). This along ith (2.5) implies the folloing. Proposition 2.1. For any eight and for any f L 1 ( ), Q Mf c f L 1, L 1.

DUAL MUCKENHOUPT-WHEEDEN PROBLEM 5 2.2. Calderón-Zygmund operators. Let K(x, y) be a locally integrable function defined off the diagonal x = y in, hich satisfies the size estimate c (2.6) K(x, y) x y n and, for some ε > 0, the regularity condition x z ε (2.7) K(x, y) K(z, y) + K(y, x) K(y, z) c x y, n+ε henever 2 x z < x y. A linear operator T : C0 ( ) L 1 loc (Rn ) is a Calderón-Zygmund operator if it extends to a bounded operator on L 2 ( ), and there is a kernel K satisfying (2.6) and (2.7) such that T f(x) = K(x, y)f(y)dy for any f C0 ( ) and x supp(f). We shall need the folloing estimate due to R. Coifman [2, 3]: for any 0 < p < and for A, (2.8) T f p dx c (Mf) p dx. The folloing theorem has been recently proved in [11]. Theorem 2.2. Let 1 < p < and let ν p = p2 p 1 log ( e + 1 p 1). There is a constant c = c(n, T ) such that for any eight, (2.9) T L p () c ν p A1. Let T be the adjoint operator of T. Then T is also a Calderón- Zygmund operator. Applying (2.9) to T instead of T and using duality e have that (2.9) is equivalent to (2.10) T L p ( 1 p ) c ν p A1, here, as usual, 1/p + 1/p = 1. 3. Proofs of main results Proof of Theorem 1.2. We start ith some ideas used in [5]. Fix p > 1. Let Sf = M( f() 1 1/2p) () 1 1/2p. Note that by (2.4) along ith (2.3), a eight () 1 r(1 1 2p ) belongs to A r ith corresponding constants independent of. Hence, by the

6 ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS PÉREZ Muckenhoupt theorem [12], S is bounded on L r for any r > 1. Therefore, by the Marcinkieicz interpolation theorem [1, p. 225], S is bounded on L p,1. We no apply the Rubio de Francia algorithm (see [9]) to define the operator R by S j h(x) Rh(x) = (2K), j here K is the norm of S on L p,1. It is easy to see that (a) h(x) Rh(x); j=0 (b) Rh L p,1 () 2 h L p,1 () ; (c) S(Rh)(x) 2KRh(x). It follos from the last property that (Rh)() 1 1/2p ith the constant bounded by 2K. Using this fact and since 1, e M 3 conclude by (2.3) that (3.1) (Rh)M (M 3 ) 1 p (Rh)()1 (M 3 ) 1 2p 1 2p A 2 ith the A 2 constant depending only on p and n. Observe no that for any p > 1 e have T f T f ( ) 1/p T f M 3 = M 3 M 3 L 1, Next, by duality, ( ) 1/p T f M 3 L p, L 1, = sup T f 1 p h p L p, dx, (M 3 ) 1 p here the supremum is taken over all non-negative h L p,1 h L p,1 = 1. Applying (3.1) and (2.8), e obtain T f 1 p h dx (M 3 ) 1 p c T f 1 p Mf(x) 1 (Rh) dx (M 3 ) 1 p 1 (Rh)()1 2p p dx. (M 3 ) 1 2p. ith

DUAL MUCKENHOUPT-WHEEDEN PROBLEM 7 From the last estimate and from (2.2) e get T f 1 ( ) 1/p p Mf h dx c (M 3 ) 1 p M 2 ( ) 1/p Mf 2c M 2 = 2c Mf M 2 1/p L 1, () Rh L L p, p (),1 () L p, () (e have used also property (b) of Rh and that h L p,1 = 1). Applying Proposition 2.1 to the last inequality completes the proof. Proof of Theorem 1.4. We follo here the classical method of Calderón- Zygmund although ith some modifications. Fix λ > 0, and set Ω λ = x : M c (f/)(x) > λ}. Let j Q j be the Whitney covering of Ω λ. Set b(x) = sup j (f f Qj )χ Qj (x) and g(x) = f(x) b(x). By (2.5), Hence, e have to estimate } T f x Ω λ : > λ (Ω λ ) c n λ f L 1. + } T b x Ω λ : > λ/2 } T g x Ω λ : > λ/2 I 1 + I 2. By [7, Ineq. (5.4), p. 92] or [9, Lemma 5, p. 413], e get I 1 2 T b(x) dx c f f Qj dx c λ \Ω λ λ Q j λ f L 1, here c = c(t, n). Applying (2.10) for 1 < p < 2 yields I 2 cpp p g p 1 p dx λ p cpp p λ p ( \Ω λ f p 1 p dx + j j ( f Qj ) p Q j 1 p dx ).

8 ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS PÉREZ We have that f c a.e. on \ Ω λ, and hence f p 1 p dx λ p 1 f L 1. \Ω λ Next, by properties of the Whitney covering, it is easy to see that for any cube Q j there exists a cube Q j such that Q j Q j, Q j c n Q j, and the center of Q j lies outside of Ω λ. Therefore, ( f Qj ) p 1 dx p 1 ( f Qj ) p 1 dx Q j 1 p ( ) p 1 f Q c p 1 j Q j c(λ A1 ) p 1 Q j, Q j hich gives ( f Qj ) p j Q j 1 p dx c(λ A1 ) p 1 j c(λ A1 ) p 1 f L 1. Combining the previous estimates, e obtain I 2 cpp 2p 1 f L 1. λ Choose no p 1 = 1 + log(e+ A1. Then e get ) I 2 c log(e + A1 ) f L 1. λ Q j () 1 p f Qj Q j This, along ith estimates for I 1 and for (Ω λ ), completes the proof. 4. Concluding remarks Remark 4.1. Folloing (2.1), e extend the definition of M k to fractional order, setting for any α 1, M α 1 ( f ) α 1dy. f(x) = sup f log + e Q x Q f Q Q It as shon in [16] that M 2 in Theorem 1.1 can be replaced by M 1+ε for any ε > 0 (ith corresponding constant depending on ε). Similarly, one can sho that M 3 in Theorem 1.2 can be replaced by M 2+ε for any ε > 0.

DUAL MUCKENHOUPT-WHEEDEN PROBLEM 9 Remark 4.2. It is easy to see that actually in Theorem 1.2 e proved a stronger inequality, namely, x : sup λ T f(x) M 3 } > λ c f dx. This yields an additional indication that Theorem 1.2 should be true ith M 2 instead of M 3. Remark 4.3. Combining ideas used in the proving Theorem 1.4 ith some estimates obtained in [16], one can sho that Theorem 1.2 can be improved by replacing M 3 by a smaller eight (M 3 ) 1 ε ε for any 0 < ε < 1/2. We emphasize, hoever, that a more principal question of interest if M 3 can be replaced by M 2 or simply by. References [1] C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, Ne York, 1988. [2] R.R. Coifman, Distribution function inequalities for singular integrals, Proc. Nat. Acad. Sci. USA, 69 (1972), 2838 2839. [3] R.R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241 250. [4] R.R. Coifman and R. Rochberg, Another characterization of BM O, Proc. Amer. Math. Soc., 79 (1980), 249 254. [5] D. Cruz-Uribe, J.M. Martell and C. Pérez, Weighted eak-type inequalities and a conjecture of Sayer, Int. Math. Res. Not., 30 (2005), 1849 1871. [6] D. Cruz-Uribe and C. Pérez, To-eight extrapolation via the maximal operator, J. Funct. Anal., 174 (2000), no. 1, 1 17. [7] J. Duoandikoetxea, Fourier Analysis, Grad. Studies Math. 29, Amer. Math. Soc., Providence, 2000. [8] C. Fefferman and E.M. Stein, Some maximal inequalities, Amer. J. Math., 93 (1971), 107 115. [9] J. García-Cuerva and J.L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. Stud. 116, North-Holland, Amsterdam, 1985. [10] T. Ianiec and G. Martin, Geometric function theory and non-linear analysis, Oxford University Press, 2001. [11] A.K. Lerner, S. Ombrosi and C. Pérez, Sharp bounds for Calderón-Zygmund operators and the relationship ith a problem of Muckenhoupt-Wheeden, preprint. Available at http://.math.biu.ac.il/ lernera/publications.html [12] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207 226. [13] B. Muckenhoupt and R. Wheeden, personal communication. [14] B. Muckenhoupt and R. Wheeden, Some eighted eak-type inequalities for the Hardy-Littleood maximal function and the Hilbert transform, Indiana Univ. Math. J. 26 (1977), 801 816.

10 ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS PÉREZ [15] S. Ombrosi y C. Pérez, Mixed eak type estimates: Examples and counterexamples related to a problem of E. Sayer, preprint. [16] C. Pérez, Weighted norm inequalities for singular integral operators, J. London Math. Soc., 49 (1994), 296 308. [17] E.T. Sayer, A eighted eak type inequality for the maximal function, Proc. Amer. Math. Soc. 93 (1985), 610 614. Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain E-mail address: aklerner@netvision.net.il Departamento de Matemática, Universidad Nacional del Sur, Bahía Blanca, 8000, Argentina Current address: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain E-mail address: sombrosi@uns.edu.ar Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain E-mail address: carlosperez@us.es