Coarse Quotient Mappings between Metric Spaces

We give a definition of coarse quotient mapping and show that several results for uniform quotient mapping also hold in the coarse setting. In particular, we prove that any Banach space that is a coarse quotient of $L_p\equiv L_p[0,1]$, $1<p<\infty$, is isomorphic to a linear quotient of $L_p$. It is also proved that $\ell_q$ is not a coarse quotient of $\ell_p$ for $1<p<q<\infty$ using Rolewicz's property ($\beta$).