The wave equation for the Bessel Laplacian

We study radial solutions of the Cauchy problem for the wave equation in the multidimensional unit ball B d , 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 L p data, the series converges in the L 2 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 L p − L 2 estimates for the solution of the heat equation and the extension problem for the fractional Bessel Laplacian are obtained.