Quantitative nonlinear embeddings into Lebesgue sequence spaces

TL;DR

This paper explores the nonlinear geometric properties of Lebesgue sequence spaces, providing solutions to embedding problems and insights into coarse embeddings, with applications in geometric group theory.

Contribution

It offers a quantitative analysis of nonlinear embeddings into Lebesgue sequence spaces and solves key embedding problems involving $ ext{ell}_q$ and $ ext{ell}_p$ spaces.

Findings

Positive solution to the strong embeddability problem from $ ext{ell}_q$ into $ ext{ell}_p$ for $0<p<q extless=1

New insights on coarse embeddability from $L_q$ into $ ext{ell}_q$, $q>2$

Exact $ ext{ell}_p$-compressions of $ ext{ell}_2$ computed

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.