Improved resolvent estimates for constant-coefficient elliptic operators in three dimensions

Abstract

We prove new $L^p$-$L^q$-estimates for solutions to elliptic differential operators with constant coefficients in ${\mathbb{R}}^3$. We use the estimates for the decay of the Fourier transform of particular surfaces in ${\mathbb{R}}^3$ with vanishing Gaussian curvature due to Erdős--Salmhofer to derive new Fourier restriction--extension estimates. These allow for constructing distributional solutions in $L^q({\mathbb{R}}^3)$ for $L^p$-data via limiting absorption by well-known means.

superseded by arXiv:2107.13400 and not intended for publication anymore