Boundedness of the Orthogonal Projection on Harmonic Fock Spaces

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