Improved bounds in the scaled Enflo type inequality for Banach Spaces

Abstract

It is shown that if (X; ∥ · ∥X) is a Banach space with Rademacher type p ≥ 1 then for every n ∈ N there exists an even integer m . ≲n2-1/p log n such that for every f : ℤₘⁿ → X, Ex;" [ ‖f ( x + m /2 ℰ ) − f(x)‖ₓₚ] .≲X mp Σn j=1 Ex [ ∥f(x + ej) − f(x)∥p X ] ; where the expectation is with respect to uniformly chosen x ∈ ℤₘⁿ and " ∈ {−1; 1}ⁿ. This improves a bounds of m ≲ n₃⁻₂/ₚ=p that was obtained in [7]. The proof is based on an augmentation of the \smoothing and approximation" scheme, which was implicit in [7].