Boundedness of the orthogonal projection on Harmonic Fock spaces

Abstract

The main result of this paper refers to the boundedness of the orthogonal projection $P_{\alpha }:L^2({\mathbb{R}}^n,d{\mu }_{\alpha })\to {\mathcal{H}}_{\alpha }^2,n\ge 2$ associated to the harmonic Fock space ${\mathcal{H}}_{\alpha }^2,$ where $d{\mu }_{\alpha }(x)=(\pi \alpha {)}^{-n/2}{e}^{-\frac{|x{|}^2}{\alpha }}dx.$ We prove that the operator $P_{\alpha }$ is not bounded on $L^p({\mathbb{R}}^n,d{\mu }_{\beta })$ when $0<p<1$ and we found a necessary and sufficient condition for the boundedness when $1\le p<\infty$ and $n$ is an even integer.

22 pages