Parseval-Type Relations and $$L^p$$ L p -Inequalities for the Operators with Complex Gaussian Kernels

Abstract

In this paper we study new \(L^p\)–boundedness properties for the integral transform with complex Gaussian kernel over \(L^p ({\mathbb {R}},(1+x^2)^{\alpha } dx)\), \(1\le p \le \infty \), \(\alpha \in {\mathbb {R}}\). We also obtain Parseval-type relations over these spaces. The Gauss–Weierstrass semigroup on \({\mathbb {R}}\) is analyzed as a particular case.