Majorizing measures and proportional subsets of bounded orthonormal systems

الأصل · EN

In this article we prove that for any orthonormal system (ⱼ)ⱼ₌₁ⁿ ⊂ L₂ that is bounded in L∞, and any 1 < k <n, there exists a subset I of cardinality greater than n-k such that on {ᵢ}ᵢ ∈ ᵢ, the L₁ norm and the L₂ norm are equivalent up to a factor μ(μ)⁵/², where μ= √n/k √ k. The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures.