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 \[ \mathcal{H}_{k,l}=(-Δ)^{l}+V(x), \] where $V$ is a strictly positive homogeneous potential of polynomial growth of order $2k$. By studying a Hörmander metric adapted to the quasi-homogeneous symbol $|ξ|^{2l}+V(x)$, as in \cite{MR4299820, MR4944933} we place $\mathcal{H}_{k,l}$ and its fractional powers within the Weyl-Hörmander calculus. In this setting, we show that the fractional operators $\mathcal{H}_{k,l}^β$, $β>0$, are globally hypoelliptic pseudodifferential operators and derive refined symbol estimates for the heat semigroup $e^{-t\mathcal{H}_{k,l}^β}$. 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 $\mathcal{H}_{k,l}^β$, 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.

Theorems & Definitions (29)

• Lemma 2.1: MR4849356
• Definition 2.2: Symbol class
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6: Hörmander metric
• Definition 2.7: $g$-weight
• Definition 2.8
• Lemma 3.1
• Lemma 3.2