The Fourier extension operator of distributions in Sobolev spaces of the sphere and the Helmholtz equation

TL;DR

This paper characterizes all entire solutions of the Helmholtz equation generated by the Fourier extension operator applied to distributions in Sobolev spaces on the sphere, providing two distinct analytical descriptions.

Contribution

It introduces two new characterizations of these solutions, one using weighted norms with spherical Laplacian powers and another resembling classical Herglotz wave functions.

Findings

Characterization in terms of $L^2$-weighted norms involving spherical Laplacian

Description of solutions via multivariable square functions for $eta>0$

Representation using fractional integral operators for $eta<0

Abstract

The purpose of this paper is to characterize all the entire solutions of the homogeneous Helmholtz equation (solutions in $\mathbb{R}^d$) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^\alpha(\mathbb{S}^{d-1}),$ with $\alpha\in \mathbb{R}$. We present two characterizations. The first one is written in terms of certain $L^2$-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For $\alpha>0$ this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for $\alpha<0$ it is written in terms of an spherical integral operator acting as a fractional integration operator. Finally, we also characterize all the solutions that are the Fourier extension operator of distributions in the sphere.