Uniform estimates for averages of order statistics of matrices

Abstract

We prove uniform estimates for the expected value of averages of order statistics of matrices in terms of their largest entries. As an application, we obtain similar probabilistic estimates for p norms via real interpolation.

Estimating Averages of Order Statistics of Bivariate Functions

“…In this work we extend our results from [16] and study averages of order statistics of bivariate functions a : {1, . .…”

“…, n}. In [16], this was established in the following setting: under some assumptions on the normalized counting measure P on a collection G of maps from I = {1, . .…”

“…where e 1 is the first standard unit vector of R n , • = • 1 , and say δ = 1/4, there exists a choice of N sign vectors that satisfy (5.4). The last ingredient is a special case of a result we recently obtained in [16,Theorem 1.4]) and reads as follows: let n ∈ N, a ∈ R n×n , and 1 ≤ p < ∞. Let G be a collection of maps from I = {1, .…”

Embeddings of Orlicz-Lorentz spaces into $L_1$

“…Readers familiar with interpolation theory might recognize expressions of this type. Indeed, a result of this flavor has also been obtained via real interpolation techniques by Lechner, Passenbrunner, and Prochno in [14].…”

“…Let us give some references most relevant to this article. It was shown by Schütt in [32, Corollary 3], using combinatorial tools partly developed in [12] (see also [14]), that every Orlicz space with a 2-concave Orlicz function embeds into L 1 . A similar result was later obtained by Lechner, Passenbrunner, and Prochno showing that a 2-concave Orlicz space ℓ n M embeds into ℓ cn 3 1 with absolute constant c ∈ (0, ∞) [15] and the embedding for 2-concave Musielak-Orlicz spaces was proved in [26,Corollary 1.3].…”