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Arithmetic, interpolation and factorization of amalgams

Tomasz Kiwerski, Jakub Tomaszewski

Abstract

Building upon Bennett's and Grosse-Erdmann's ideas falling under the conceptual umbrella of factorization of inequalities, we propose a unified approach towards the structure of certain Banach ideal spaces defined in terms of the least decreasing majorant. The key to our results, and, it seems, the main novelty in general, is the synthesis of discretization process, usually called the blocking technique, along with some tools from the interpolation theory. This blend allows us to obtain an abstract versions of several remarkable results proposed by Bennett and to show certain phenomena in new, somehow more complete perspective. Furthermore, with the help of technology we have developed, we re-prove and sometimes also improve many more recent results belonging to this circle of ideas.

Arithmetic, interpolation and factorization of amalgams

Abstract

Building upon Bennett's and Grosse-Erdmann's ideas falling under the conceptual umbrella of factorization of inequalities, we propose a unified approach towards the structure of certain Banach ideal spaces defined in terms of the least decreasing majorant. The key to our results, and, it seems, the main novelty in general, is the synthesis of discretization process, usually called the blocking technique, along with some tools from the interpolation theory. This blend allows us to obtain an abstract versions of several remarkable results proposed by Bennett and to show certain phenomena in new, somehow more complete perspective. Furthermore, with the help of technology we have developed, we re-prove and sometimes also improve many more recent results belonging to this circle of ideas.
Paper Structure (50 sections, 44 theorems, 414 equations, 4 tables)

This paper contains 50 sections, 44 theorems, 414 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Motivation and goal
  3. Results
  4. Overview of the key ideas
  5. Statements and Declarations
  6. Acknowledgments
  7. Funding
  8. Conflict of Interest
  9. Data Availibility
  10. Toolbox
  11. Notation and terminology
  12. Foreground roles
  13. Kalton's zone
  14. Boyd indices
  15. Elements on interpolation spaces
  16. ...and 35 more sections

Key Result

Theorem A

For any three Banach ideal spaces, say $E$, $F$ and $G$, with $G$ having the Fatou property, the following holds

Theorems & Definitions (170)

  • Remark 1.c.1
  • Remark 1.c.3: Wiener amalgam spaces
  • Remark 1.c.4
  • Example 1.c.5: S. V. Astashkin and L. Maligranda, 2009
  • Remark 1.c.6: About Example
  • Remark 3.d.2
  • proof
  • Remark 3.e.1
  • proof
  • Remark 3.f.1: Sequence spaces
  • ...and 160 more