A simple proof of a theorem of Jean Bourgain

1991/06/03 by Gilles Pisier, Pisier, Gilles · 1 citation
Mathematics · #46E #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #math.FA #msc:46E

paper · pdf · doi:10.48550/arxiv.math/9201228

arxiv created 1991/06/03 · openalex publication_date 1991/06/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a simple proof of Bourgain's disc algebra version of Grothendieck's theorem, i.e. that every operator on the disc algebra with values in L₁ or L₂ is 2-absolutely summing and hence extends to an operator defined on the whole of C. This implies Bourgain's result that L₁/H¹ is of cotype 2. We also prove more generally that L_r/H^r is of cotype 2 for 0<r< 1.

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