The gaussian convolution and reproducing kernels associated with the Hankel multidimensional operator

We consider the Hankel multidimensional operator defined on]0, +∞[ⁿ by

$${\Delta _\alpha} = \sum\limits_{j = 1}^n {\left({{{{\partial ^2}} \over {\partial x_j^2}} + {{2{\alpha _j} + 1} \over {{x_j}}}{\partial \over {\partial {x_j}}}} \right)} $$

where \(\alpha = ({\alpha _1},{\alpha _2}, \ldots ,{\alpha _n}) \in ] - {1 \over 2}, + \infty {[^n}\). We give the most important harmonic analysis results related to the operator Δ_α (translation operators τ_x, convolution product * and Hankel transform ℌ_α).

Using harmonic analysis results, we study spaces of Sobolev type for which we make explicit kernels reproducing. Next, we define and study the gaussian convolution \({{\cal G}^t}\), t > 0, associated with the Hankel multidimensinal operator Δ_α. This transformation generalizes the classical gaussian transformation. We establish the most important properties of this transformation. In particular, we show that the gaussian transformation solves the heat equation, that is

$${\Delta _\alpha}(u)(x,t) = {{\partial u} \over {\partial t}}(x,t),\,\,\,\,\,(x,t) \in [0, + \infty {[^n} \times ]0, + \infty [.$$

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