Characterizations of a class of Pilipovi{ć} spaces by powers of harmonic oscillator

Abstract

We show that a smooth function $f$ on ${\mathbf{R}}^d$ belongs to the Pilipovi{ć} space ${\mathcal{H}}_{{♭}_{\sigma }}({\mathbf{R}}^d)$ or the Pilipovi{ć} space ${\mathcal{H}}_{0,{♭}_{\sigma }}({\mathbf{R}}^d)$, if and only if the $L^p$ norm of $H_d^{N}f$ for $N\ge 0$, satisfy certain types of estimates. Here $H_d=|x{|}^2-{\Delta }_x$ is the harmonic oscillator.

13 pages. This is the third version. Some mathematical details have been clarified in this version compared to earlier versions