Majorizing measures and their applications

Date of Completion

January 1999






Majorizing measure techniques are developed and applied to Banach space theory. In particular, the following is proved. ^ Let B1 and B2 be the unit balls of ln1 and ln2 , respectively, relative to the canonical basis ei ni=1 . Suppose K⊂logp nB1∩B 2 . Then for every 3 > 0, there exist S⊂1,2,&ldots;,n with cardinality n1-3 , and constant C depending only on 3 and p, such that K∩YS⊂CB1∩YS , where YS is the linear span of eii∈S . ^ The following is a consequence. ^ Consider vectors x1,x2,&ldots;,xn in the unit ball of a Banach space X, and s=supi≤ nx* xi 2;x*∈X*,∥ x* ∥ ≤1. If X is of type 2 and X* is uniformly convex, then, there exists a constant C depending only on 3 and T2X , such that for a randomly selected subset I of cardinality m=n1-3/s , i∈I aixi≤C i∈I ai2 1/2, for all scalar sequence ai ni=1 . ^ This solves a problem stated in [T2]. ^