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 ${\mathbb R}^n$ which has the property that there exists a $t > 1/(p-1)$ such that $$ \sup\limits _{0 < r\leq 1} \int \limits_{{\mathbb R}^n} \Big({\mu (B(x,r))\over r^p}\Big)^t{\rm d}\mu (x) < \infty . $$ We show that, for such a $\mu $, any singular integral operator on $L^2({\mathbb R}^n,\mu )$ with a smooth, homogeneous kernel of degree $-1$ belongs to the norm ideal ${\cal C}^+_{p/(p-1)}$.

Keywords: Singular integral operator, norm ideal

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