Non-existence of reflectionless measures for the s-Riesz transform when 0<s<1

Abstract

A measure $\mu$ on ${\mathbb{R}}^d$ is called reflectionless for the $s$-Riesz transform if the singular integral $R^s\mu (x)=\int \frac{y-x}{|y-x{|}^{s+1}}d\mu (y)$ is constant on the support of $\mu$ in some weak sense and, moreover, the operator defined by $R_{\mu }^s(f)=R^s(f\mu )$ is bounded in $L^2(\mu )$. In this paper we show that the only reflectionless measure for the $s$-Riesz transform is the zero measure when $0<s<1$.

11 pages