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

In this paper, we study an extension problem for the Ornstein-Uhlenbeck operator $L=-\mathrm{\Delta }+2x\cdot \mathrm{\nabla }+n$ 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.