The Poisson Convolution Associated with the Spherical Mean Operator

Abstract

The spherical mean operator $\mathscr {R}$ is defined on the space of continuous functions on $\mathbb {R}\times \mathbb {R}^n,$ even with respect to the first variable by

$$\begin{aligned} \mathscr {R}(f)(r, x)=\int _{S^n} f\big ((0, x)+r \omega \big )d\sigma (\omega ) \end{aligned}$$

where $S^n$ is the unit sphere of $\mathbb {R}\times \mathbb {R}^n,$ and $d\sigma$ is the euclidian measure on $S^n,$ normalized to have total mass 1. We study the most important properties of harmonic analysis related to the spherical mean operator (translation operators, convolution product and Fourier transform). Using harmonic analysis results, we study spaces of Sobolev type for which we make explicit kernels reproducing. Next, we define and study the Poisson convolution $\mathscr {P}_t, t>0,$ associated with the spherical mean operator $\mathscr {R}.$ We establish the most important properties of the Poisson convolution. In particular, we show that the Poisson convolution solves the wave equation, namely

$$\begin{aligned} \** (u)(r, x, t)= - \frac{\partial ^2 u}{\partial t^2}(r, x, t),\ (r, x, t)\in \mathbb {R}\times \mathbb {R}^n\times ]0, +\infty [, \end{aligned}$$

where $\**$ is the Laplacian,

$$\begin{aligned} \** = \frac{\partial ^2}{\partial r^2}+\frac{n}{r}\ \frac{\partial }{\partial r}+\sum _{j=1}^n \frac{\partial ^2}{\partial x_j^2}. \end{aligned}$$

In the second part of this work, we prove the existence and uniqueness of the extremal function associated with the Poisson convolution. We express this extremal function using the reproducing kernels and we establish important estimates for this function.