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 $R^s_\mu(f)=R^s(f\,\mu)$ 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 $0<s<1$.