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Propagation of Shubin-Sobolev singularities of Weyl-quantizations of complex quadratic forms

Marcello Malagutti, Alberto Parmeggiani, Davide Tramontana

TL;DR

This work extends Hörmander's microlocal theory to the isotropic Shubin framework and studies how Shubin-Sobolev singularities propagate under Weyl-quantized quadratic generators. By developing the s-isotropic wave front set $WF_{\mathrm{iso}}^s$ and connecting it to time-frequency concepts via Weyl–Wick and Gabor analysis, the authors establish microlocal propagation estimates along the imaginary part of the symbol, constrained by the singular space $S$ and governed by the Hamilton map $F$. A central result shows $WF_{\mathrm{iso}}^{s-\mu}(e^{-tA}u_0) \subseteq (e^{t H_{a_I}}(WF_{\mathrm{iso}}^{s}(u_0) \cap S)) \cap S$ with a derivative loss up to $4n+\varepsilon$, and kernel regularity $K_{e^{-2itF}} \in B^{-(n+\varepsilon)}$ provides quantitative control. The framework unifies isotropic microlocal analysis with time-frequency techniques, yields robust propagation descriptions for heat-like and quantum harmonic oscillator-type evolutions, and offers tools for analyzing more general complex quadratic Weyl-quantized generators in $\mathbb{R}^n$.

Abstract

The aim of this work is to develop the Hörmander microlocal theory in the isotropic framework and use the results we obtain to study the propagation of singularities for an evolution problem, with diffusive part given by a Weyl-quantization of a complex quadratic form on the phase space.

Propagation of Shubin-Sobolev singularities of Weyl-quantizations of complex quadratic forms

TL;DR

This work extends Hörmander's microlocal theory to the isotropic Shubin framework and studies how Shubin-Sobolev singularities propagate under Weyl-quantized quadratic generators. By developing the s-isotropic wave front set and connecting it to time-frequency concepts via Weyl–Wick and Gabor analysis, the authors establish microlocal propagation estimates along the imaginary part of the symbol, constrained by the singular space and governed by the Hamilton map . A central result shows with a derivative loss up to , and kernel regularity provides quantitative control. The framework unifies isotropic microlocal analysis with time-frequency techniques, yields robust propagation descriptions for heat-like and quantum harmonic oscillator-type evolutions, and offers tools for analyzing more general complex quadratic Weyl-quantized generators in .

Abstract

The aim of this work is to develop the Hörmander microlocal theory in the isotropic framework and use the results we obtain to study the propagation of singularities for an evolution problem, with diffusive part given by a Weyl-quantization of a complex quadratic form on the phase space.

Paper Structure

This paper contains 7 sections, 41 theorems, 384 equations.

Table of Contents

  1. Introduction
  2. Preliminaries of isotropic calculus
  3. Isotropic microsingularities
  4. Relation with the time-frequency calculus
  5. Main properties of the $s$-isotropic wave front set
  6. Propagation of Shubin-Sobolev singularities
  7. Auxiliary results

Key Result

Theorem 1.1

Let $\mathscr{K}:\mathscr{S}(\mathbb{R}^n)\rightarrow \mathscr{S}'(\mathbb{R}^m)$, with $n \geq m$, be a linear operator with Schwartz kernel $K \in B^{-r_1}(\mathbb{R}^{m+n})$ for $r_1\geq 0$. Then, if $u \in \mathscr{S}(\mathbb{R}^n)$ and $\mu>n$ is fixed, we have that for all $s_1 \in \mathbb{R}$ In addition, the definition of the map $u \mapsto \mathscr{K}u$ can be extended by continuity to th

Theorems & Definitions (112)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Definition 2.8
  • ...and 102 more