Bourgain discretization using Lebesgue-Bochner spaces

Abstract

We study the Lebesgue-Bochner discretization property of Banach spaces Y , which ensures that the Bourgain discretization modulus for Y has a good lower estimate. We prove that there exist spaces that do not have the Lebesgue-Bochner discretization property, and we give a class of examples of spaces that enjoy this property.

Key words: Bourgain discretization theorem, distortion of an embedding, Lebesgue-Bochner space.