The inverse, the heat semigroup, Liouville’s theorems and the spectrum for the Grushin operator

Abstract

By decomposing the Grushin operator on \({\mathbb{R}2}\) into a family of parameterized Hermite operators, we give estimates for the inverses and the heat semigroups of these Hermite operators, which are then used to obtain Sobolev estimates for the inverse and the heat semigroup of the Grushin operator. Using the global hypoellipticity of the parametrized Hermite operators, Liouville’s theorems for harmonic functions of the Grushin operator on \({\mathbb{R}2}\) are obtained. The spectrum of the Grushin operator is computed.