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 $L_p\equiv L_p[0,1]$, $1<p<\mathrm{\infty }$, is isomorphic to a linear quotient of $L_p$. It is also proved that ${\mathrm{\ell }}_q$ is not a coarse quotient of ${\mathrm{\ell }}_p$ for $1<p<q<\mathrm{\infty }$ using Rolewicz’s property ($\beta$).