On the smoothness of eigenfunctions of hyponormal singular integral operators

Abstract:

Fix $\varphi \in {L^\infty }(E)$ and let $E \subset R$ be bounded and measurable; for $1 < p < \infty$ consider the bounded linear operator \[ Tf(s) = sf(s) + \frac {{\varphi (s)}}{\pi }\int _E^\ast \frac {{\bar \varphi (t)f(t)}}{{t - s}}dt\;{\mkern 1mu} {\text {a}}{\text {.e}}{\text {. }}s \in E\] where $f \in {L^p}(E)$. If $\nu = \lambda + i\mu \in C$ then there are no nonzero ${L^p}(E)$ solutions of $Tf = \nu f$ for $p > 2$ in case $\lambda$ is a point of positive Lebesgue density in the complement of $E$.