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Trapasso, S. Ivan
Wave packet analysis of semigroups generated by quadratic differential operators Journal Article
In: J. Differential Equations, vol. 449, iss. 2025, pp. 113683, 2025.
Abstract | Links | BibTeX | Tags: partial differential equations, pseudodifferential operators, semigroups, time-frequency analysis, wave packets
@article{T_JDE_25,
title = {Wave packet analysis of semigroups generated by quadratic differential operators},
author = {S. Ivan Trapasso},
url = {https://arxiv.org/abs/2408.11130, arXiv
https://www.sciencedirect.com/science/article/pii/S0022039625007107/pdfft?md5=aacb48f12ad8e134791022b7e499449c&pid=1-s2.0-S0022039625007107-main.pdf, PDF},
doi = {10.1016/j.jde.2025.113683},
year = {2025},
date = {2025-08-02},
urldate = {2025-08-02},
journal = {J. Differential Equations},
volume = {449},
issue = {2025},
pages = {113683},
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.},
keywords = {partial differential equations, pseudodifferential operators, semigroups, time-frequency analysis, wave packets},
pubstate = {published},
tppubtype = {article}
}
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.