An extension problem, trace Hardy and Hardy's inequalities for Ornstein-Uhlenbeck operator

TL;DR

This paper extends the analysis of the Ornstein-Uhlenbeck operator by developing an extension problem, deriving trace Hardy inequalities, and establishing new estimates for fractional powers, advancing understanding of its functional inequalities.

Contribution

It introduces a novel extension problem for the Ornstein-Uhlenbeck operator, leading to new trace Hardy inequalities and $L^p-L^q$ estimates for fractional powers.

Findings

Derived a trace Hardy inequality for the Ornstein-Uhlenbeck operator.

Established an isometry property of the solution operator.

Obtained new $L^p-L^q$ estimates for fractional Hermite operators.

Abstract

In this paper, we study an extension problem for the Ornstein-Uhlenbeck operator $L=-\Delta+2x\cdot\nabla +n$ and we obtain various characterisations of the solution of the same. We use a particular solution of that extension problem to prove a trace Hardy inequality for $L$ from which Hardy's inequality for fractional powers of $L$ is obtained. We also prove an isometry property of the solution operator associated to the extension problem. Moreover, new $L^p-L^q$ estimates are obtained for the fractional powers of the Hermite operator.