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March, 2003 Maximal functions and singular integrals associated to polynomial mappings of $\mathbb{R}^n$
Anthony Carbery, Fulvio Ricci, James Wright
Rev. Mat. Iberoamericana 19(1): 1-22 (March, 2003).

Abstract

We consider convolution operators on $\mathbb{R}^n$ of the form $$T_Pf(x) =\int_{\mathbb{R}^m} f\big(x-P(y)\big)K(y) dy$$, where $P$ is a polynomial defined on $\mathbb{R}^m$ with values in $\mathbb{R}^n$ and $K$ is a smooth Calderón-Zygmund kernel on $\mathbb{R}^m$. A maximal operator $M_P$ can be constructed in a similar fashion. We discuss weak-type 1-1 estimates for $T_P$ and $M_P$ and the uniformity of such estimates with respect to $P$. We also obtain $L^p$-estimates for "supermaximal" operators, defined by taking suprema over $P$ ranging in certain classes of polynomials of bounded degree.

Citation

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Anthony Carbery. Fulvio Ricci. James Wright. "Maximal functions and singular integrals associated to polynomial mappings of $\mathbb{R}^n$." Rev. Mat. Iberoamericana 19 (1) 1 - 22, March, 2003.

Information

Published: March, 2003
First available in Project Euclid: 31 March 2003

zbMATH: 1036.42013
MathSciNet: MR1993413

Subjects:
Primary: 42B20, 42B25

Rights: Copyright © 2003 Departamento de Matemáticas, Universidad Autónoma de Madrid

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