Wave packet analysis of semigroups generated by quadratic differential operators

TL;DR

This paper conducts a phase space analysis of semigroups generated by quadratic differential operators, providing bounds for wave packet decompositions and extending regularity results to modulation spaces.

Contribution

It introduces new pointwise bounds for Gabor wave packet coefficients and extends phase regularity analysis of quadratic semigroups to all modulation spaces.

Findings

Derived pointwise bounds for matrix coefficients of wave packet decompositions.

Extended phase regularity results from $L^2$ to all modulation spaces $M^p$.

Provided explicit bounds for the semigroup behavior in phase space.

Abstract

We perform a phase space analysis of evolution equations associated with the Weyl quantization $q^{\mathrm{w}}$ of a complex quadratic form $q$ on $\mathbb{R}^{2d}$ with non-positive real part. In particular, we obtain pointwise bounds for the matrix coefficients of the Gabor wave packet decomposition of the generated semigroup $e^{tq^{\mathrm{w}}}$ if $\mathrm{Re} (q) \le 0$ and the companion singular space associated is trivial. This result is then leveraged to achieve a comprehensive analysis of the phase regularity of $e^{tq^{\mathrm{w}}}$ with $\mathrm{Re} (q) \le 0$, thereby extending the $L^2$ analysis of quadratic semigroups initiated by Hitrik and Pravda-Starov to general modulation spaces $M^p(\mathbb{R}^d)$, $1 \le p \le \infty$, with optimal explicit bounds.