Phase-Space Analysis of generalised Fractional Anharmonic and Ornstein-Uhlenbeck Semigroups on Weighted Modulation Spaces

TL;DR

This paper develops a phase-space framework for fractional anharmonic oscillators and Ornstein-Uhlenbeck semigroups, establishing their boundedness, smoothing properties, and well-posedness of related nonlinear heat equations on weighted modulation spaces.

Contribution

It introduces a phase-space approach for fractional anharmonic and Ornstein-Uhlenbeck operators, deriving symbol estimates and proving their boundedness and smoothing effects on modulation spaces.

Findings

Fractional operators are globally hypoelliptic pseudodifferential operators.

Heat semigroups are bounded and smoothing on weighted modulation spaces.

Global well-posedness results for nonlinear heat equations with these operators.

Abstract

We develop a phase-space framework for fractional generalised anharmonic oscillators and their heat semigroups on weighted modulation spaces. We consider operators of the form \[ \mathcal{H}_{k,l}=(-\Delta)^{l}+V(x), \] where $V$ is a strictly positive homogeneous potential of polynomial growth of order $2k$. By studying a H\"ormander metric adapted to the quasi-homogeneous symbol $|\xi|^{2l}+V(x)$, as in \cite{MR4299820, MR4944933} we place $\mathcal{H}_{k,l}$ and its fractional powers within the Weyl-H\"ormander calculus. In this setting, we show that the fractional operators $\mathcal{H}_{k,l}^{\beta}$, $\beta>0$, are globally hypoelliptic pseudodifferential operators and derive refined symbol estimates for the heat semigroup $e^{-t\mathcal{H}_{k,l}^{\beta}}$. These estimates yield boundedness and smoothing properties of the fractional anharmonic heat semigroup on weighted modulation spaces $\mathcal{M}^{p,q}_{s}$, for the full range $0<p,q\leq\infty$ and suitable range of $s$. As applications, we establish global well-posedness of nonlinear heat equations associated with $\mathcal{H}_{k,l}^{\beta}$, including both homogenous power and spatially inhomogenous nonlinearities. Finally, we introduce Gaussian modulation spaces adapted to the Ornstein-Uhlenbeck operator and prove continuity of the corresponding semigroup, providing a phase-space perspective complementary to classical Gaussian harmonic analysis.