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Besov spaces and Schatten class Hankel operators for Hardy and Paley--Wiener spaces in higher dimensions

Konstantinos Bampouras, Karl-Mikael Perfekt

TL;DR

The paper develops a unified framework connecting Schatten-class Hankel operators on Paley--Wiener spaces to Besov regularity adapted to domain geometry. By introducing geometry-aware weights $\omega_{\Omega}$ and Besov spaces $B_{p,p}^{s}(2\Omega)$ built from admissible parallelepiped covers, it characterizes Hankel operator membership $\mathop{H}_{\varphi}^{\sigma,\tau}\in S^{p}$ (for $1\le p\le 2$) precisely as $\varphi\in B_{p,p}^{\sigma+\tau+\frac{1}{p}}(2\Omega)$, with extensions to all $1\le p<\infty$ on product Hardy spaces and to $p<2(n+1)/(n-1)$ for bounded smooth domains. The work combines Macbeath-region geometry, kernel estimates, and interpolation to yield a robust, geometry-aware Schatten-class theory for high-dimensional Hankel operators, including special cases for polytopes and product spaces. It also raises open questions about admissibility and convolution-type inequalities for the weight $\omega_{\Omega}$, guiding future development of Paley--Wiener Besov theory in irregular domains.

Abstract

We consider Schatten class membership of Hankel operators on Paley--Wiener spaces of convex $Ω\subset \mathbb{R}^n$, both for bounded and unbounded domains. In particular, the classical product Hardy spaces fit within our theory. For admissible domains, we develop a framework and theory of Besov spaces of Paley--Wiener type, and prove that a Hankel operator belongs to the Schatten class $S^p$ if and only if its symbol belongs to a corresponding Besov space, for $1 \leq p \leq 2$. We extend this result to all $1 \leq p < \infty$ for the classical product Hardy spaces and to $1 \leq p < 2(n+1)/(n-1)$ for the Paley--Wiener space of a bounded smooth domain $Ω\subset \mathbb{R}^n$ of strictly positive curvature.

Besov spaces and Schatten class Hankel operators for Hardy and Paley--Wiener spaces in higher dimensions

TL;DR

The paper develops a unified framework connecting Schatten-class Hankel operators on Paley--Wiener spaces to Besov regularity adapted to domain geometry. By introducing geometry-aware weights and Besov spaces built from admissible parallelepiped covers, it characterizes Hankel operator membership (for ) precisely as , with extensions to all on product Hardy spaces and to for bounded smooth domains. The work combines Macbeath-region geometry, kernel estimates, and interpolation to yield a robust, geometry-aware Schatten-class theory for high-dimensional Hankel operators, including special cases for polytopes and product spaces. It also raises open questions about admissibility and convolution-type inequalities for the weight , guiding future development of Paley--Wiener Besov theory in irregular domains.

Abstract

We consider Schatten class membership of Hankel operators on Paley--Wiener spaces of convex , both for bounded and unbounded domains. In particular, the classical product Hardy spaces fit within our theory. For admissible domains, we develop a framework and theory of Besov spaces of Paley--Wiener type, and prove that a Hankel operator belongs to the Schatten class if and only if its symbol belongs to a corresponding Besov space, for . We extend this result to all for the classical product Hardy spaces and to for the Paley--Wiener space of a bounded smooth domain of strictly positive curvature.
Paper Structure (7 sections, 41 theorems, 183 equations, 1 figure)

This paper contains 7 sections, 41 theorems, 183 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Besov spaces of Paley--Wiener type
  3. Sufficient conditions for trace class and boundedness
  4. Necessity of the Besov space condition
  5. Smooth domains in $\mathbb{R}^n$ with positive curvature
  6. Hankel operators on the product Hardy space
  7. Open Problems and Further Directions

Key Result

Theorem 1.1

Let $1\leq p<\infty$ and $\sigma,\tau>\max(-\frac{1}{2},-\frac{1}{p})$. The extended Hankel operator $\mathop{\mathrm{H}}\nolimits_\varphi^{\sigma, \tau} \colon H^2(\mathbb{R} \times \cdots \times \mathbb{R}) \to H^2(\mathbb{R} \times \cdots \times \mathbb{R})$ belongs to the Schatten class $S^{p}$

Figures (1)

  • Figure :

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 65 more