Characterization of solutions of a generalized Helmholtz problem

Abstract

In this article, we classify all distributional solutions of $f(-\hat{I}”)u=f(1)u$ where $f$ is a non-constant Bernstein function. Specifically, we show that the Fourier transform of $u$ is a single-layer distribution on the unit sphere. Examples of such operators include $(-\hat{I}”{)}^{\ddot{I}}ƒ$ (for $\ddot{I}ƒ\in (0,1]$), $\log (1-\hat{I}”)$ and $(-\hat{I}”{)}^{\frac{1}{2}}\text{tanh}((-\hat{I}”{)}^{\frac{1}{2}})$.