Localization for general Helmholtz

TL;DR

This paper generalizes the equivalence between classical and fractional Helmholtz equations, providing a broad framework that reduces the problem to analyzing the support of the Fourier symbol under mild conditions.

Contribution

It introduces a new, general setup for Helmholtz equivalence, extending previous results to a wider class of symbols with minimal assumptions.

Findings

Established a broad framework for Helmholtz equivalence

Reduced the problem to support localization of Fourier symbols

Applicable to symbols satisfying mild regularity conditions

Abstract

In \cite{gmw2022}, Guan, Murugan and Wei established the equivalence of the classical Helmholtz equation with a ``fractional Helmholtz" equation in which the Laplacian operator is replaced by the nonlocal fractional Laplacian operator. More general equivalence results are obtained for symbols which are complete Bernstein and satisfy additional regularity conditions. In this work we introduce a novel and general set-up for this Helmholtz equivalence problem. We show that under very mild and easy-to-check conditions on the Fourier multiplier, the general Helmholtz equation can be effectively reduced to a localization statement on the support of the symbol.