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 $associatedtotheharmonicFockspace$$\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 $1\le p<\mathrm{\infty }$ and $n$ is an even integer.