NOTES ON COMPACTNESS IN Lp-SPACES ON LOCALLY COMPACT GROUPS

Abstract

The main goal of the paper is to provide new insight into compactness in $L^p$-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 “$L^p$-properties” ($L^p$-boundedness, $L^p$-equicontinuity, and $L^p$-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 $L^p$-spaces on locally compact groups.