On Operators with Complex Gaussian Kernels over $L^p$ spaces

Abstract

In this paper we study new $L^p$-boundedness properties and Parseval-type relations concerning the operators with complex Gaussian kernels over the spaces $L^{p}(\mathbb{R},w(x) dx)$, $1\leq p \leq \infty$, where $w$ represents any function greater than or equal to one almost everywhere on $\mathbb{R}$. Here, the Gauss-Weierstrass semigroup is considered as a particular case.