Boundedness of the Orthogonal Projection on Harmonic Fock Spaces

Abstract

The main result of this paper refers to the boundedness of the orthogonal projection $P_{\alpha }:L^{2}({\mathbb {R}}^{n},d\mu _{\alpha })\rightarrow {\mathcal {H}}_{\alpha }^{2}, n\ge 2$ 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.