Pointwise convergence of fractional powers of Hermite type operators

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 $R>0$.

arXiv comment19 pages. This paper is dedicated to the memory of Eleonor (Pola) Harboure. Revised version, to appear in Rev Unión Matem Argentina