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

TL;DR

This paper establishes new $L^p$-$L^q$ estimates for elliptic operators with constant coefficients in three dimensions, utilizing Fourier restriction techniques to improve resolvent bounds and solution regularity.

Contribution

It introduces novel Fourier restriction-extension estimates based on decay properties of surfaces with vanishing Gaussian curvature, leading to improved resolvent estimates.

Findings

Derived new $L^p$-$L^q$ estimates for elliptic operators in $\

Constructed distributional solutions in $L^q(\

Enhanced understanding of Fourier transform decay on specific surfaces.

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 Erd\H{o}s--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.