Natural proof of the characterization of relatively compact families in\n L^p-spaces on locally compact groups

2018/01/05 by Mateusz Krukowski, Krukowski, Mateusz
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1801.01898

openalex publication_date 2018/01/05 · openalex created_date 2022/09/03 · openalex updated_date 2026/07/28

Abstract

In the paper we look for an elegant proof of the characterization of\nrelatively compact families in L^p-spaces. At first glance, the suggested\napproach may seem convoluted and lengthy, but we spare no effort to argue that\nour proof is in fact more natural than the ones existent in the literature. The\nkey idea is that the three properties which characterize relative compactness\nin L^p-spaces (L^p-boundedness, L^p-equicontinuity and\nL^p-equivanishing) are "preserved" (or rather "inherited") when the family\n Ffamily\⊂ L^p is convolved with a continuous and compactly supported\nfunction. This new family turns out to be relatively compact in C₀-space and\nit remains to be demonstrated that relative compactness in C₀-space implies\nrelative compactness of the original family Ffamily.\n

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