[Paper Review] Continuity and Schatten properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces or Hörmander classes

Joachim Toft|arXiv (Cornell University)|Jun 15, 2014

Mathematical Analysis and Transform Methods23 references4 citations

TL;DR

This paper establishes continuity and Schatten-von Neumann properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces or Hörmander classes, extending prior results to include Lebesgue and Schatten parameters in (0, ∞]. Using Gabor analysis and matrix operator factorization, it proves that operators in Schatten-von Neumann class $\mathscr{I}_p$ for $p \in (0, \infty]$ are characterized by symbol decay in $L^p$-type norms, even when $p < 1$, overcoming challenges from non-local convexity in these spaces.

ABSTRACT

We establish continuity and Schatten-von Neumann properties for matrix operators with matrices satisfying mixed quasi-norm estimates. These considerations also include the case when the Lebesgue and Schatten parameters are allowed to stay between $0$ and $1$. We use the results to deduce continuity and Schatten-von Neumann properties for pseudo-differential operators with symbols in a broad class of modulation spaces.

Motivation & Objective

• To extend Schatten-von Neumann and continuity properties of pseudo-differential operators to the full range $p \in (0, \infty]$, including the non-locally convex case $p < 1$.
• To overcome technical challenges arising from the failure of local convexity in quasi-Banach modulation spaces and $L^p$ spaces with $p < 1$.
• To generalize known results for $p \geq 1$ to $p < 1$ by developing new matrix operator estimates with mixed quasi-norms.
• To establish a link between symbol decay in $L^p$-type norms and membership in Schatten-von Neumann classes $\mathscr{I}_p$ for $p \in (0, \infty]$.
• To provide a comprehensive framework for continuity and compactness of pseudo-differential operators using Gabor analysis and factorization techniques in non-locally convex settings.

Proposed method

• Derives continuity and Schatten-von Neumann properties for matrix operators with entries satisfying mixed quasi-norm estimates in $\ell^p$ and $\ell^q$ spaces with $p, q \in (0, \infty]$.
• Applies Gabor analysis to represent pseudo-differential operators via time-frequency localization, enabling transfer of matrix results to operator classes.
• Uses factorization techniques for matrix operators to handle non-local convexity in quasi-Banach spaces, particularly for $p < 1$.
• Establishes a connection between the symbol's decay in $L^p$-type norms and the singular value decay of the operator, leveraging the equivalence $\operatorname{Op}(a) \in \mathscr{I}_p \Leftrightarrow a \in L^p$ for Hörmander classes.
• Applies weighted estimates and modulation space norms with $p < 1$ by using $v$-moderate weights and properties of the short-time Fourier transform.
• Proves that $\|a*b\|_{L^p} \lesssim \|a\|_{s_t^p} \|b\|_{s_t^p}$ for $p \in (0,1]$, extending convolution estimates to the quasi-Banach setting.

Experimental results

Research questions

• RQ1 Can Schatten-von Neumann properties for pseudo-differential operators be extended to the case $p < 1$, where standard local convexity fails?
• RQ2 How can continuity and compactness of pseudo-differential operators be characterized when symbols belong to quasi-Banach modulation spaces with $p < 1$?
• RQ3 What matrix operator estimates with mixed quasi-norms are sufficient to deduce Schatten-von Neumann membership for pseudo-differential operators?
• RQ4 To what extent can Gabor analysis and factorization techniques be adapted to non-locally convex function spaces to study pseudo-differential operators?
• RQ5 Is the equivalence $\operatorname{Op}(a) \in \mathscr{I}_p \Leftrightarrow a \in L^p$ valid for $p \in (0, \infty]$ when $a$ belongs to a Hörmander class or modulation space?

Key findings

• For $p \in (0,1]$, the convolution map $(a,b) \mapsto a*b$ is continuous from $s_t^p \times s_t^p$ to $L^p$, extending classical results to the quasi-Banach setting.
• The symbol space $s_t^p$ embeds continuously into $\mathscr{F}L^p$, the Fourier image of $L^p$, for $p \in (0, \infty]$, even when $p < 1$.
• The operator norm of a pseudo-differential operator $\operatorname{Op}(a)$ satisfies $\|\operatorname{Op}(a)\|_{\mathscr{I}_p} \lesssim \|a\|_{s_t^p}$ for $p \in (0, \infty]$, generalizing known results to $p < 1$.
• For $a \in M^{p,p}$ with $p \in (0,2]$, the operator $\operatorname{Op}(a)$ belongs to $\mathscr{I}_p$, extending the classical result $\operatorname{Op}(a) \in \mathscr{I}_p \Leftrightarrow a \in L^p$ to $p < 1$.
• The embedding $s_t^p \hookrightarrow \mathscr{F}L^p$ holds for $p \in (0, \infty]$, which implies that symbol decay in $s_t^p$ implies singular value decay in $\ell^p$.
• The result $\|a*b\|_{L^p} \lesssim \|a\|_{s_t^p} \|b\|_{s_t^p}$ for $p \in (0,1]$ is established via $L^{2p}$-norm estimates on short-time Fourier transforms and Cauchy-Schwarz inequalities, enabling the extension to $p < 1$.