# Boundedness of the orthogonal projection on Harmonic Fock spaces

## TL;DR

This paper investigates the boundedness of the orthogonal projection on harmonic Fock spaces, establishing conditions for boundedness across different L^p spaces and showing unboundedness for p<1.

## Contribution

It provides a complete characterization of the boundedness of the orthogonal projection on harmonic Fock spaces for various p and dimensions, including necessary and sufficient conditions.

## Key findings

- Projection not bounded on L^p for 0<p<1
- Necessary and sufficient conditions for boundedness when p≥1
- Boundedness depends on the dimension being an even integer

## 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\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.