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.

Mathematics Subject Classification (2010):

• Primary: 46B85
• Secondary: 46B06, 46B07, 46E40

Key words:

• Bourgain discretization theorem
• distortion of an embedding
• Lebesgue-Bochner space