# Characterizations of a class of Pilipovi{\'c} spaces by powers of harmonic oscillator

## TL;DR

This paper characterizes Pilipovi{\'c} spaces of smooth functions on \(\mathbf{R}^d\) using growth estimates of powers of the harmonic oscillator applied to the functions.

## Contribution

It provides a new characterization of Pilipovi{\'c} spaces via $L^p$ norm estimates of harmonic oscillator powers, linking functional space membership to operator estimates.

## Key findings

- Characterization of Pilipovi{\'c} spaces through harmonic oscillator norms.
- Equivalence between space membership and specific $L^p$ estimates.
- Provides criteria for smooth functions based on operator growth conditions.

## Abstract

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