Quantitative nonlinear embeddings into Lebesgue sequence spaces

Abstract

In this paper fundamental nonlinear geometries of Lebesgue sequence spaces are studied in their quantitative aspects. Applications of this work are a positive solution to the strong embeddability problem from $\ell_q$ into $\ell_p$ ($0<p<q\le 1$) and new insights on the coarse embedabbility problem from $L_q$ into $\ell_q$, $q>2$. Relevant to geometric group theory purposes, the exact $\ell_p$-compressions of $\ell_2$ are computed. Finally coarse deformation of metric spaces with property A and locally compact amenable groups is investigated.Comment: 34 page

Tight Embeddability of Proper and Stable Metric Spaces

“…It follows from Kalton and Randrianarivony [18] that is not almost Lipschitzly embeddable into p when p ∈ [ , ∞) and p ≠ . However it was shown by the rst author [2] that α p ( ) = for p ∈ [ , ).…”

Metric Inequalities

“…where in (14), as well as in (15), (16), (17) and (18) below, the expectation is with respect to ε ∈ {−1, 1} n chosen uniformly at random. Since, by Jensen's inequality,…”

“…The analogue of Conjecture 1.8 for sequence spaces has a positive answer. Indeed, a combination of [14,Cor. 2.19] and [14,Cor.…”

“…Indeed, a combination of [14,Cor. 2.19] and [14,Cor. 2.23] shows that for every 1 q p < ∞, if θ ∈ (0, 1] is such that the metric space (ℓ q , x − y θ q ) admits a bi-Lipschitz embedding into ℓ p then necessarily θ q/p.…”

On the equivalence between coarse and uniform embeddability of quasi-Banach spaces into a Hilbert space

“…Baudier [Ba,Corollaries 2.23 and 2.19] proved that if 0 < p < q < ∞ and q ≥ 1, then (2) s ℓq (ℓ p ) = α ℓq (ℓ p ) = max{p, 1} q (the case q = 1 was already proved in [Al,Proposition 4.1(ii)…”