2023 NOTES ON COMPACTNESS IN Lp-SPACES ON LOCALLY COMPACT GROUPS
Mateusz Krukowski
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Publ. Mat. 67(2): 687-713 (2023). DOI: 10.5565/PUBLMAT6722308

Abstract

The main goal of the paper is to provide new insight into compactness in Lp-spaces on locally compact groups. The article begins with a brief historical overview and the current state of literature regarding the topic. Subsequently, we “take a step back” and investigate the Arzelà–Ascoli theorem on a non-compact domain together with one-point compactification. The main idea comes in Section 3, where we introduce the “Lp-properties” (Lp-boundedness, Lp-equicontinuity, and Lp-equivanishing) and study their “behaviour under convolution”. The paper proceeds with an analysis of Young’s convolution inequality, which plays a vital role in the final section. During the “grand finale”, all the pieces of the puzzle are brought together as we lay down a new approach to compactness in Lp-spaces on locally compact groups.

Acknowledgements

First and foremost, I would like to express my deepest gratitude towards both anonymous reviewers, whose patient study of the paper, insightful remarks, and shrewd comments allowed for a tremendous improvement of the original version of my work. I cannot overestimate the impact that their contribution had on the quality of the article.

Next, I wish to thank Wojciech Kryszewski, whose penetrating questions forced me to repeatedly rethink the ideas that I have been working on. It is a privilege to have such a great source of constructive criticism. I also wish to thank Robert Stańczy for the reference to the Arzelà–Ascoli theorem for C0(X) in [8].

Citation

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Mateusz Krukowski. "NOTES ON COMPACTNESS IN Lp-SPACES ON LOCALLY COMPACT GROUPS." Publ. Mat. 67 (2) 687 - 713, 2023. https://doi.org/10.5565/PUBLMAT6722308

Information

Received: 7 May 2021; Revised: 21 October 2021; Published: 2023
First available in Project Euclid: 29 June 2023

MathSciNet: MR4609016
zbMATH: 07720475
Digital Object Identifier: 10.5565/PUBLMAT6722308

Subjects:
Primary: 43A15
Secondary: 46B50 , ‎46E15 , 46E30

Keywords: Arzelà–Ascoli theorem , Kolmogorov–Riesz theorem , Sudakov theorem , Weil theorem , Young’s convolution inequality

Rights: Copyright © 2023 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.67 • No. 2 • 2023
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