Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Compactness of Fourier concentration operators

Helge Jørgen Samuelsen

TL;DR

The paper establishes a sufficient condition for the compactness of Fourier concentration operators $Q_FP_E$ on $L^2({\mathbb R}^d)$ by introducing the concept of sets that are very thin at infinity. It proves that if both $E$ and $F$ are very thin at infinity, then $Q_FP_E$ is compact, leveraging a Littlewood–Paley decomposition together with the Logvinenko–Sereda and Shubin–Vakilian–Wolff uncertainty principles. This approach provides a partial answer to Katsnelson and Machluf's question on truncated Fourier operators, and clarifies how the behavior of $E$ and $F$ at infinity governs operator compactness. The work also highlights the role of cross-term control in decomposed operator analysis and extends the understanding of concentration phenomena in harmonic analysis.

Abstract

We present a sufficient condition on sets $E$ and $F$ in $\mathbb{R}^d$ to ensure compactness of Fourier concentration operators by introducing the notion of sets which are very thin at infinity. We are able to show that if the sets $E$ and $F$ are both very thin at infinity, then the associated Fourier concentration operator is compact on $L^2(\mathbb{R}^d)$. The proof relies on a combination of the Logvinenko-Sereda uncertainty principle together with an uncertainty principle due to Shubin, Vakilian and Wolff. This provides a partial answer to a question posed by Katsnelson and Machluf on truncated Fourier operators.

Compactness of Fourier concentration operators

TL;DR

The paper establishes a sufficient condition for the compactness of Fourier concentration operators on by introducing the concept of sets that are very thin at infinity. It proves that if both and are very thin at infinity, then is compact, leveraging a Littlewood–Paley decomposition together with the Logvinenko–Sereda and Shubin–Vakilian–Wolff uncertainty principles. This approach provides a partial answer to Katsnelson and Machluf's question on truncated Fourier operators, and clarifies how the behavior of and at infinity governs operator compactness. The work also highlights the role of cross-term control in decomposed operator analysis and extends the understanding of concentration phenomena in harmonic analysis.

Abstract

We present a sufficient condition on sets and in to ensure compactness of Fourier concentration operators by introducing the notion of sets which are very thin at infinity. We are able to show that if the sets and are both very thin at infinity, then the associated Fourier concentration operator is compact on . The proof relies on a combination of the Logvinenko-Sereda uncertainty principle together with an uncertainty principle due to Shubin, Vakilian and Wolff. This provides a partial answer to a question posed by Katsnelson and Machluf on truncated Fourier operators.

Paper Structure

This paper contains 4 sections, 9 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Uncertainty Principles and proof of Theorem \ref{['thm:MainThm']}
  3. Decomposition of the identity
  4. Logvinenko-Sereda Uncertainty Principle

Key Result

Theorem 1.1

There are $\varepsilon>0$ and $C<\infty$ such that if $E,F\subseteq {\mathbb R}^d$ are $\varepsilon$-thin sets, then for any $f\in L^2({\mathbb R}^d)$

Theorems & Definitions (16)

  • Theorem 1.1: Wolff_Uncertainty, Thm $2.1$
  • Definition 1.2: Very thin at infinity
  • Theorem 1.3
  • Lemma 2.1: Wolff_Uncertainty, Lem. $2.2$
  • Proposition 2.2
  • proof
  • Proposition 2.3: Schlag, Prop $10.6$
  • Lemma 2.4
  • proof
  • Corollary 2.4.1
  • ...and 6 more