On weighted inequalities for singular integrals

Abstract

In this note we consider singular integrals associated to CalderonZygmund kernels. We prove that if the kernel is supported in (0, oo) then the one-sided Ap condition, A-, is a sufficient condition for the singular integral to be bounded in LP(w), 1 < p < oo, or from Ll(wdx) into weak-Ll(wdx) if p = 1. This one-sided Ap condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in (0, oo). The two-sided version of this result is also obtained: Muckenhoupt's Ap condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calder6n-Zygmund kernel which is not the function zero either in (-oo,cO) or in (0, oo). INTRODUCTION It is a classical result in the theory of weighted inequalities the fact that the Ap condition of B. Muckenhoupt on a weight w is equivalent to the LP(wdx) boundedness of the Hilbert transform. This result was proved in 1973 by Hunt, Muckenhoupt and Wheeden [HMW]. In 1974 Coifman and Fefferman [CF] gave a different proof which relies on a good-A inequality, producing an integral estimate of the singular integral in terms of the Hardy-Littlewood maximal operator. Since 1986 the work by E. Sawyer [S], Andersen and Sawyer [AS], Martin Reyes, Ortega Salvador and de la Torre [MOT], [MT] has shown that many positive operators of real analysis have one-sided versions for which the classes of weights are larger than Muckenhoupt's ones. Our purpose here is to study the corresponding problems for singular integrals. The situation for one-sided singular integrals is different. The symmetry properties of the Hilbert kernel produce the necessary cancellation properties of a singular integral, so that, no one-sided truncation of l/x is expected to produce a one-sided singular integral. Nevertheless, as we show in Lemma (1.5), the class of general singular integral Calderon-Zygmund kernels supported on a half line is nontrivial. We ask for the more general class of weights w for which such singular integral operators are bounded in LP(wdx). It turns out (Theorem (2.1)) that the one-sided Ap condition is a sufficient condition which becomes also necessary when we require Received by the editors March 15, 1995 and, in revised form, January 30, 1996. 1991 Mathematics Subject Classification. Primary 42B25.