Mathematical Research Letters

Volume 23 (2016)

Number 6

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

Pages: 1665 – 1679



Michael Goldberg (Department of Mathematical Sciences, University of Cincinnati, Ohio, U.S.A.)


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.

Accepted 31 August 2015

Published 21 February 2017