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

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

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