The wave equation for the Bessel Laplacian

Abstract

We study radial solutions of the Cauchy problem for the wave equation in the multidimensional unit ball Bd, d≥1. In this case, the operator that appears is the Bessel Laplacian and the solution u(t,x) is given in terms of a Fourier–Bessel expansion. We prove that, for initial Lp data, the series converges in the L2 norm. The analysis of a particular operator, the adjoint of the Riesz transform for Fourier–Bessel series, is needed for our purposes, and may be of independent interest. As applications, certain Lp−L2 estimates for the solution of the heat equation and the extension problem for the fractional Bessel Laplacian are obtained.