Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On weighted inequalities for singular integrals

Authors: H. Aimar, L. Forzani and F. J. Martín-Reyes
Journal: Proc. Amer. Math. Soc. 125 (1997), 2057-2064
MSC (1991): Primary 42B25
MathSciNet review: 1376747
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Abstract: In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in $(0,\infty )$ then the one-sided $A_{p}$ condition, $A_{p}^{-}$, is a sufficient condition for the singular integral to be bounded in $L^{p}(w)$, $1<p<\infty $, or from $L^{1}(wdx)$ into weak-$L^{1}(wdx)$ if $p=1$. This one-sided $A_{p}$ 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 $(0,\infty )$. The two-sided version of this result is also obtained: Muckenhoupts $A_{p}$ condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in $(-\infty ,0)$ or in $(0,\infty )$.

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Additional Information

H. Aimar
Affiliation: Dept. Matematica, FIQ, Prop.CAI+D, INTEC, Gëmes 3450, 3000 Santa Fe, Argentina

F. J. Martín-Reyes
Affiliation: Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain

Keywords: Singular integrals, Calderon-Zygmund operators, weights
Received by editor(s): March 15, 1995
Received by editor(s) in revised form: January 30, 1996
Additional Notes: The research of the third author has been partially supported by D.G.I.C.Y.T. grant (PB91-0413) and Junta de Andalucía.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1997 American Mathematical Society