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# Quantitative nonlinear embeddings into Lebesgue sequence spaces

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 $ℓq$ into $ℓp$ ($0) and new insights on the coarse embeddability problem from $Lq$ into $ℓq$, $q>2$. Relevant to geometric group theory purposes, the exact $ℓp$-compression of $ℓ2$ is computed. Finally coarse deformation of metric spaces with property A and locally compact amenable groups is investigated.

AMSC: 46B20, 46B85, 46T99, 20F65

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