Journal of Operator Theory

Volume 49, Issue 2, Spring 2003 pp. 311-324.

Singular integral operators associated with measures of varying density

Authors: Jingbo Xia
Author institution: Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, USA

Summary: Let

$1 < p < \infty$ and let

$\mu$ be a compactly supported regular Borel measure on

${\bbb R}^n$ which has the property that there exists a

$t > 1/(p-1)$ such that

$\sup\limits _{0<r\leq 1} \int \limits_{{\bbb R}^n} \Big({\mu (B(x,r))\over r^p}\Big)^t\d\mu (x) < \infty .$ We show that, for such a

$\mu$ , any singular integral operator on

$L^2({\bbb R}^n,\mu )$ with a smooth, homogeneous kernel of degree

$-1$ belongs to the norm ideal

${\cal C}^+_{p/(p-1)}$ .

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