Pointwise convergence of fractional powers of Hermite type operators

TL;DR

This paper establishes minimal conditions for pointwise convergence of fractional powers of Hermite and Ornstein-Uhlenbeck operators, and extends results to related fractional operators, highlighting optimality through examples.

Contribution

It provides the first minimal integrability and smoothness conditions for pointwise definitions of fractional Hermite and Ornstein-Uhlenbeck operators, and extends findings to related fractional operators.

Findings

Identified minimal conditions for pointwise fractional Hermite operator convergence

Proved optimality of conditions with various examples

Extended results to fractional operators with added potential R

Abstract

When $L$ is the Hermite or the Ornstein-Uhlenbeck operator, we find minimal integrability and smoothness conditions on a function $f$ so that the fractional power $L^\sigma f(x_0)$ is well-defined at a given point $x_0$. We illustrate the optimality of the conditions with various examples. Finally, we obtain similar results for the fractional operators $(-\Delta+R)^\sigma$, with $R>0$.