Images of some subspaces of $$L^2(\mathbb {R}^{m})$$ L 2 ( R m ) under Grushin and Hermite semigroup

Abstract

The image of a subspace of \(L^2(\mathbb {R}^{n+1})\) under Grushin semigroup is characterized as direct sum of two weighted Bergman spaces. Hermite Sobolev space of positive order on \(\mathbb {R}^n\) is defined, and the image of this space under Hermite semigroup is found out using Caputo fractional derivative as weighted Bergman space. The connection of Grushin operator and parametrized Hermite operator is used in this paper extensively and with that help image of Grushin Sobolev space of positive order is characterized.