The main goal of the paper is to provide new insight into compactness in -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 “-properties” (-boundedness, -equicontinuity, and -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 -spaces on locally compact groups.
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 in .
"NOTES ON COMPACTNESS IN -SPACES ON LOCALLY COMPACT GROUPS." Publ. Mat. 67 (2) 687 - 713, 2023. https://doi.org/10.5565/PUBLMAT6722308