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

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.

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Accepted 31 August 2015

Published 21 February 2017