Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A unified approach to inequalities for K-functionals and moduli of smoothness

Amiran Gogatishvili, Bohumir Opic, Sergey Tikhonov, Walter Trebels

Abstract

The paper provides a detailed study of crucial inequalities for smoothness and interpolation characteristics in rearrangement invariant Banach function spaces. We present a unified approach based on Holmstedt formulas to obtain these estimates. As examples, we derive new inequalities for moduli of smoothness and K-functionals in various Lorentz spaces.

A unified approach to inequalities for K-functionals and moduli of smoothness

Abstract

The paper provides a detailed study of crucial inequalities for smoothness and interpolation characteristics in rearrangement invariant Banach function spaces. We present a unified approach based on Holmstedt formulas to obtain these estimates. As examples, we derive new inequalities for moduli of smoothness and K-functionals in various Lorentz spaces.
Paper Structure (14 sections, 22 theorems, 266 equations)

This paper contains 14 sections, 22 theorems, 266 equations.

Table of Contents

  1. Introduction
  2. Some basic results
  3. Inequalities for moduli of smoothness on Lorentz spaces
  4. Remarks and proofs in outlines
  5. Sharp Ul'yanov and Kolyada inequalities for $p=1$
  6. Notation and preliminaries
  7. General inequalities for K-functionals
  8. Holmstedt-type formulas
  9. Inequalities for $K$-functionals involving the potential-type operators
  10. The Ul'yanov inequality between weighted Lorentz spaces
  11. Definitions and preliminaries
  12. The Ul'yanov inequality between weighted Lorentz spaces.
  13. The Ul'yanov inequality for Lorentz-Karamata spaces: a first look.
  14. Sharp Ul'yanov inequality between the Lorentz--Karamata spaces

Key Result

Lemma 3.1

If the function $\xi$ given above is left continuous on $(a,b)$, then and Moreover, if $\xi\in C((a,b)),$ then

Theorems & Definitions (49)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • Theorem 4.1
  • Remark 4.2
  • proof : Proof of Theorem \ref{['holmsted-f']}
  • Theorem 4.3
  • Remark 4.4
  • ...and 39 more