Log-Sobolev inequalities for semi-direct product operators and applications

TL;DR

This paper establishes logarithmic Sobolev inequalities for semi-direct product operators, demonstrating their applications in ultracontractive bounds of semigroups and Hardy's inequalities, especially for Grushin-type operators.

Contribution

It introduces new logarithmic Sobolev inequalities for semi-direct product operators and applies them to various operator classes and semigroup bounds.

Findings

Proved logarithmic Sobolev inequalities for semi-direct product operators

Applied inequalities to ultracontractive bounds of semigroups

Established Hardy's inequalities for Grushin-type operators

Abstract

We prove logarithmic Sobolev inequalities for semi-direct product operators (see definition in Section 1). We apply our main results to examples of operators and provide some applications to ultracontractive bounds of semigroups. Hardy's inequalities are proved and use to study some operators of Grushin's type.