Nonlinear Quotients of Banach Spaces

Abstract

We study quotients of Banach spaces in three nonlinear categories: Lipschitz, uniform and coarse. Following a brief review of what has been known for uniform and Lipschitz quotients of classical Banach spaces, we introduce the definition of coarse quotient and show that several results for uniform quotients also hold in the coarse setting. In particular, we prove that any Banach space that is a coarse quotient of Lp = Lp[0, 1], 1 < p < (infinity), is isomorphic to a linear quotient of Lp. It is also proven, by applying a geometric notion of Rolewicz called property (B), that lq is not a coarse quotient of lp for 1 < p < q < (infinity), and c0 is not a coarse quotient of any Banach space with property (B). On the other hand, we give a sharp distortion lower bound for embedding the countably branching tree into a Banach space with property (B). It is then shown how this work unifies and extends a series of results in the nonlinear quotient theory of Banach space.