On weighted inequalities for singular integrals
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- by H. Aimar, L. Forzani and F. J. Martín-Reyes PDF
- Proc. Amer. Math. Soc. 125 (1997), 2057-2064 Request permission
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: Muckenhoupt’s $A_{p}$ 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 $(0,\infty )$.References
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Additional Information
- H. Aimar
- Affiliation: Dept. Matematica, FIQ, Prop.CAI+D, INTEC, Gëmes 3450, 3000 Santa Fe, Argentina
- Email: haimar@fiqus.unl.edu.ar
- F. J. Martín-Reyes
- Affiliation: Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
- Email: martin_reyes@ccuma.uma.es
- 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
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2057-2064
- MSC (1991): Primary 42B25
- DOI: https://doi.org/10.1090/S0002-9939-97-03787-8
- MathSciNet review: 1376747