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Maximal noncompactness of embeddings into Marcinkiewicz spaces

Jan Malý, Zdeněk Mihula, Vít Musil, Luboš Pick

TL;DR

The paper develops a universal principle for estimating the extent of noncompactness of operators into Marcinkiewicz spaces, showing that nontrivial embeddings are typically maximally noncompact via the ball measure of noncompactness $\alpha(T)$. It proves that Marcinkiewicz spaces largely fail disjoint superadditivity, yet provides abstract lower bounds (Theorems 'general-lower' and 'maximal_noncomp_alt_m_phi') that yield maximal noncompactness for a broad class of embeddings, including Sobolev-type ones with shrinking concentration. The results extend to embeddings into the space of essentially bounded functions, with a novel lower-bound technique for $L^{\infty}$ targets. Collectively, the work sharpens the threshold between compact and noncompact behavior in limiting Sobolev-type embeddings and related operator embeddings, complementing and expanding prior findings in the literature.

Abstract

We develop a new functional-analytic technique for investigating the degree of noncompactness of an operator defined on a quasinormed space and taking values in a Marcinkiewicz space. The main result is a general principle from which it can be derived that such operators are almost always maximally noncompact in the sense that their ball measure of noncompactness coincides with their operator norm. We point out specifications of the universal principle to the case of the identity operator.

Maximal noncompactness of embeddings into Marcinkiewicz spaces

TL;DR

The paper develops a universal principle for estimating the extent of noncompactness of operators into Marcinkiewicz spaces, showing that nontrivial embeddings are typically maximally noncompact via the ball measure of noncompactness . It proves that Marcinkiewicz spaces largely fail disjoint superadditivity, yet provides abstract lower bounds (Theorems 'general-lower' and 'maximal_noncomp_alt_m_phi') that yield maximal noncompactness for a broad class of embeddings, including Sobolev-type ones with shrinking concentration. The results extend to embeddings into the space of essentially bounded functions, with a novel lower-bound technique for targets. Collectively, the work sharpens the threshold between compact and noncompact behavior in limiting Sobolev-type embeddings and related operator embeddings, complementing and expanding prior findings in the literature.

Abstract

We develop a new functional-analytic technique for investigating the degree of noncompactness of an operator defined on a quasinormed space and taking values in a Marcinkiewicz space. The main result is a general principle from which it can be derived that such operators are almost always maximally noncompact in the sense that their ball measure of noncompactness coincides with their operator norm. We point out specifications of the universal principle to the case of the identity operator.
Paper Structure (9 sections, 8 theorems, 149 equations)

This paper contains 9 sections, 8 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Marcinkiewicz spaces
  3. Noncompactness
  4. Sobolev embeddings
  5. Disjoint superadditivity
  6. Overview of the main results
  7. Lack of disjoint superadditivity of Marcinkiewicz spaces
  8. Maximal noncompactness of embeddings into a Marcinkiewicz space
  9. Embeddings into the space of essentially bounded functions

Key Result

Theorem 2.1

Let $\varphi\colon(0,\mu(\mathcal{R}))\to(0,\infty)$ be a nondecreasing function satisfying $\Delta_2$ condition near zero. Then the space $m_\varphi(\mathcal{R}, \mu)$ is not disjointly superadditive.

Theorems & Definitions (12)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Definition
  • Corollary 3.4
  • Theorem 3.5
  • ...and 2 more