# Weighted inequalities for commutators of fractional and singular integrals

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## Abstract

Publicacions Matemátiques, Vol 35 (1991, 209-235 . WEIGHTED INEQUALITIES FOR COMMUTATORS OF FRACTIONAL AND SINGULAR INTEGRALS CARLOS SEGOVIA AND JOSÉ L . TORREA Introduction We dedicate this paper to the memory of José Luis Rubio de Francia, who de- veloped the theory of extrapolation and gave beautiful applications of vectorial methods in harmonic analysis . Through this paper we shall work on Rn, endowed with the Lebesgue mea- sure . Given a Banach space E we shall denote by LE(Rn) or LE the Bochner- Lebesgue space of E-valued strongly measurable functions such that where lif(x)1IE <-Foo . Given a positive measurable function ce(x) we shall denote by LÉ(a) the space of E-valued strongly measurable functions such that f 11f(x)IJÉ-(x)dx < o0 and we shall denote by BMOE(a) the space of strongly measurable functions b such that ~Q 11 b(x) - bQ11 Edx < C ~Q a(x)dx, bQ = jQ1-1 IQ b(x)dx . Given two Banach spaces E and F, we shall denote by .C(E, F) the Banach space of all continuous linear operators from E into F . By a Banach lattice we mean a partially ordered Banach space F over the reals such that (i) x < y implies x + z < y -}- z for every x, y, z E F, (ii) ax>0foreveryx>0inFanda>0inR . (iii) for every x, y E F there exists a least upper bound (l.u.b .) and a greatest lower bound (g .1 .b .), and (iv) if Ix1 is defined as ,xl = l.u.b . (x, -x) then IIxjj < llyl) whenever Ix1 < jyj . 210 C . SEGOVIA, J .L . TORREA We shall say that a positive function a belongs to A(p, q) if (1 a-P'(,)d,)1IP'(1 aq(x)dx) l I9 < C, IQl Q IQI e holds for any cube Q C Rn and p' + p = p'p, the constant C not depending on Q. Observe that if we denote by Ap the Muckenhoupt's class, then, for p > 1, w E A(p, p) if and only if wP E Ap . Finally we shall say that a Banach space E is U.M.D . if the Hilbert transform is bounded from LÉ into LÉ, see [2] . The paper is organized as follows : in section 1 we state and prove the extrap- olation results, in s

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