Resumen de Notes on compactness in Lp-spaces on locally compact groups

Mateusz Krukowski

• 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 Arzel`a–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.