Type and cotype of some Banach spaces

Mieczyslaw Mastylo

doi:10.1155/s0161171292000309

Abstract

Type and cotype are computed for Banach spaces generated by some positive sublinear operators and Banach function spaces. Applications of the results yield that under certain assumptions Clarkson's inequalities hold in these spaces.

Highlights

Given a Banach space X, we let for any n !1, the smallest constants for which p s 2 q < oo and s s < o, Ko’.)(X) and Kt,.s)(X be for every choice of xi}i.l C X, where {r" }’. denotes the sequence of Rademacher functions defined by r,(t) sign sin 2"tr for 0 s 1
If s 1, we say that X is of cotype q
For example Maurey and Pisier [8] showed that a Banach space X is of type p for some p >

Summary

Introduction

Given a Banach space X, we let for any n !1, the smallest constants for which p s 2 q < oo and s s < o, Ko’.)(X) and Kt,.s)(X be for every choice of xi}i.l C X, where {r" }’. denotes the sequence of Rademacher functions defined by r,(t) sign sin 2"tr for 0 s 1. One of the great advantages of the classification of Banach spaces in terms of type and cotype is the existence of a rather satisfactory geometric characterization of these notions. For example Maurey and Pisier [8] showed that a Banach space X is of type p for some p > The well-known examples of Banach spaces for which the above inequalities hold are L,-spaces

Results

Conclusion