Fourier-extension estimates for symmetric functions and applications to nonlinear Helmholtz equations

TL;DR

This paper develops weighted Fourier-extension estimates for symmetric functions on spheres, extending to subgroups of orthogonal groups, and applies these results to establish existence of symmetric solutions to nonlinear Helmholtz equations.

Contribution

It introduces new weighted Fourier-extension estimates for symmetric functions and applies them to prove existence of symmetric solutions to nonlinear Helmholtz equations.

Findings

Weighted Fourier-extension estimates below Stein-Tomas exponent.

Boundedness and nonvanishing of weighted Helmholtz resolvent.

Existence of G-invariant solutions to nonlinear Helmholtz equations.

Abstract

We establish weighted $L^p$-Fourier-extension estimates for $O(N-k) \times O(k)$-invariant functions defined on the unit sphere $\mathbb{S}^{N-1}$, allowing for exponents $p$ below the Stein-Tomas critical exponent $\frac{2(N+1)}{N-1}$. Moreover, in the more general setting of an arbitrary closed subgroup $G \subset O(N)$ and $G$-invariant functions, we study the implications of weighted Fourier-extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for $G$-invariant solutions to the nonlinear Helmholtz equation $$ - \Delta u - u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}(\mathbb{R}^{N}), $$ where $Q$ is a nonnegative bounded and $G$-invariant weight function.