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

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 Schrödinger 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.

10 pages

Cited by in corpus (3)

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