Fourier-extension estimates for symmetric functions and applications 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 $$-\Delta u-u=Q(x)|u{|}^{p-2}u,u\in W^{2,p}({\mathbb{R}}^N),$$ where $Q$ is a nonnegative bounded and $G$-invariant weight function.