The Helmholtz equation with $L^p$ data and Bochner-Riesz multipliers

TL;DR

This paper establishes existence results for Helmholtz equation solutions with $L^p$ data and improves Bochner-Riesz multiplier bounds on functions vanishing on the sphere, linking to the Limiting Absorption Principle.

Contribution

It introduces new existence results for Helmholtz solutions with $L^p$ data and enhances Bochner-Riesz bounds for functions with Fourier support avoiding the sphere.

Findings

Existence of $L^2$ solutions under specific $L^p$ conditions.

Improved $L^p o L^q$ bounds for Bochner-Riesz multipliers.

Connection to the Limiting Absorption Principle for Schrödinger operators.

Abstract

We prove the existence of $L^2$ solutions to the Helmholtz equation $(-\Delta - 1)u = f$ in ${\mathbb R}^n$ assuming the given data $f$ belongs to $L^{(2n+2)/(n+5)}({\mathbb R}^n)$ and satisfies the "Fredholm condition" that $\hat{f}$ vanishes on the unit sphere. This problem, and similar results for the perturbed Helmholtz equation $(-\Delta -1)u = -Vu + f$, are connected to the Limiting Absorption Principle for Schr\"odinger operators. The same techniques are then used to prove that a wide range of $L^p \mapsto L^q$ bounds for Bochner-Riesz multipliers are improved if one considers their action on the closed subspace of functions whose Fourier transform vanishes on the unit sphere.