Nonlinear weakly sequentially continuous embeddings between Banach spaces

Abstract: In these notes, we study nonlinear embeddings between Banach spaces which are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$ by a weakly sequentially continuous map, then every spreading model $(e_n)n$ of a normalized weakly null sequence in $X$ satisfies [|e_1+\ldots+e_k|{\overline{\delta}_Y}\lesssim|e_1+\ldots+e_k|_S,] where $\overline{\delta}_Y$ is the modulus of asymptotic uniform convexity of $Y$. Among other results, we obtain Banach spaces $X$ and $Y$ so that $X$ coarsely (resp. uniformly) embeds into $Y$, but so that $X$ cannot be mapped into $Y$ by a weakly sequentially continuous coarse (resp. uniform) embedding.