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

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.