Improved bounds in the scaled Enflo type inequality for Banach spaces

Details Improved bounds in the scaled Enflo type inequality for Banach spaces Improved bounds in the scaled Enflo type inequality for Banach spaces

2010-04-23

arxiv.org/abs/1004.4221v1

Mathematics

Functional Analysis

Scientific paper

It is shown that if (X,||.||_X) is a Banach space with Rademacher type p \ge 1, then for every integer n there exists an even integer m < Cn^{2-1/p}log n (C is an absolute constant), such that for every f:Z_m^n --> X, \Avg_{x,\e}[||f(x+ m\e/2)-f(x)}||_X^p] < C(p,X) m^p\sum_{j=1}^n\Avg_x[||f(x+e_j)-f(x)||_X^p], where the expectation is with respect to uniformly chosen x \in Z_m^n and \e \in \{-1,1\}^n, and C(p,X) is a constant that depends on p and the Rademacher type constant of X. This improves a bound of m < Cn^{3-2/p} that was obtained in [Mendel, Naor 2007]. The proof is based on an augmentation of the "smoothing and approximation" scheme, which was implicit in [Mendel, Naor 2007].

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