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

TL;DR

This paper proves that for the $s$-Riesz transform with $0<s<1$, the only measure that is reflectionless (constant in a weak sense and bounded in $L^2$) is the zero measure, indicating no non-trivial reflectionless measures exist.

Contribution

The paper establishes the non-existence of non-trivial reflectionless measures for the $s$-Riesz transform when $0<s<1$, filling a gap in the understanding of these transforms.

Findings

No non-zero reflectionless measures for $0<s<1$.

Reflectionless measures must be zero, implying uniqueness.

Advances the theory of singular integrals and geometric measure theory.

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 $0<s<1$.