On weighted inequalities for singular integrals
| Autor: | Liliana Forzani, H. Aimar, Francisco J. Martín-Reyes |
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| Rok vydání: | 1997 |
| Předmět: | |
| Zdroj: | Scopus-Elsevier |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-97-03787-8 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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