Boundedness of the orthogonal projection on Harmonic Fock spaces

The main result of this paper refers to the boundedness of the orthogonal projection $P_{\alpha}:L^{2}(\mathbb{R}^{n},d\mu_{\alpha})\rightarrow \mathcal{H}_{\alpha}^{2}, n\geq2$ 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 $1\leq p<\infty$ and $n$ is an even integer.