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Weighted Norm Inequalities for Fractional Integral Operators With Rough Kernel

Published online by Cambridge University Press:  20 November 2018

Yong Ding  and
Shanzhen Lu
Yong Ding
Affiliation:
Department of Mathematics Beijing Normal UniversityBeijing 100875 China, e-mail: lusz@sun.ihep.ac.cn
Shanzhen Lu
Affiliation:
Department of Mathematics Beijing Normal UniversityBeijing 100875 China, e-mail: lusz@sun.ihep.ac.cn
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Abstract

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Given function $\Omega $ on ${{\mathbb{R}}^{n}}$, we define the fractional maximal operator and the fractional integral operator by

$${{M}_{\Omega ,\,\alpha }}f(x)\,=\,_{r>0}^{\sup }\frac{1}{{{r}^{n-\alpha }}}\,\int{{{_{|y|}}_{<r}}\,}|\Omega (y)|\,|f(x-y)|\,dy$$
and
$${{T}_{\Omega ,\,\alpha }}f(x)\,=\,\int{_{{{\mathbb{R}}^{n}}}}\,\frac{\Omega (y)}{{{\left| y \right|}^{n-\alpha }}}f(x-y)dy$$
 respectively, where $0\,<\,\alpha \,<\,n$. In this paper we study the weighted norm inequalities of ${{M}_{\Omega ,\,\alpha }}$ and ${{T}_{\Omega ,\,\alpha }}$ for appropriate $\alpha ,\,s$ and $A(p,\,\,q)$ weights in the case that $\Omega \,\in \,{{L}^{s}}({{S}^{n-1}})(s>1)$, homogeneous of degree zero.

Keywords

42B2042B25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 1 , 01 February 1998 , pp. 29 - 39
Copyright
Copyright © Canadian Mathematical Society 1998

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