Stability of a pair of Banach spaces for coarse Lipschitz embeddings

Abstract

A pair of Banach spaces (X, Y) is said to be coarsely stable if for every coarse Lipschitz embedding $$f: X\rightarrow Y$$, there exist $$\alpha ,\gamma >0$$ and a Lipschitz mapping $$T:L(f)\rightarrow X$$ with its Lipschitz constant $$\Vert T\Vert _{Lip}\le \alpha $$ such that $$\Vert Tf(x)-x\Vert \le \gamma $$ for all $$x\in X$$, where L(f) is the closed linear span of f(X). In this paper, we study the coarse stability of a pair of Banach spaces (X, Y) when X is an absolute Lipschitz retract (resp. X is an arbitrary Banach space; $$L_2$$)and Y is an arbitrary Banach space (resp. Y is a Hilbert space; $$L_p$$ for $$2<p<+\infty $$).