Maximal Noncompactness of Wiener-Hopf Operators

TL;DR

The paper extends the theory of maximal noncompactness to Wiener-Hopf operators on broad classes of translation-invariant Banach function spaces. It proves that for Fourier multipliers a, the truncated Wiener-Hopf operator W(a) acting on X(R_+) is maximally noncompact, with the operator norm, essential norm, and Hausdorff measure of noncompactness all equal. The authors develop a general abstract result for translation-invariant operators and exploit density and invariance properties to obtain precise noncompactness equalities, then apply these to rearrangement-invariant spaces, including nonseparable cases such as L^{p,∞}. This yields new equalities even in the classical L^p setting and broadens the applicability to a wide family of function spaces via Zippin indices and RI structure.

Abstract

Let $X(\mathbb{R})$ be a separable translation-invariant Banach function space and $a$ be a Fourier multiplier on $X(\mathbb{R})$. We prove that the Wiener-Hopf operator $W(a)$ with symbol $a$ is maximally noncompact on the space $X(\mathbb{R}_+)$, that is, its Hausdorff measure of noncompactness, its essential norm and its norm are all equal. This equality for the Hausdorff measure of noncompactness of $W(a)$ is new even in the case of $X(\mathbb{R})=L^p(\mathbb{R})$ with $1\le p<\infty$.

Theorems & Definitions (11)

• Theorem 1.1
• Remark 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof