For a map \(f:(X,d_X)\to (Y,d_Y)\) between metric spaces let \[ \rho_f(t)=\inf\{d_Y(f(x),f(y)) : d_X(x,y)\geq t\}, \] and \[ \omega_f(t)=\sup\{d_Y(f(x),f(y)) : d_X(x,y)\leq t\}. \] When \(X\) and \(Y\) are not uniformly discrete \(f\) is called a uniform embedding if \(\omega_f(t)\to 0\) when \(t\to 0\) and \(\rho_f(t)>0\) for all \(t>0\). If \(X\) and \(Y\) are unbounded, \(f\) is called a coarse embedding if \(\rho_f(t)\to \infty\) when \(t\to \infty\) and \(\omega_f(t)<\infty\) for all \(t>0\). A strong embedding is an embedding which is simultaneously coarse and uniform.
In this paper these notions are quantified in terms of growth and relative growth of the functions \(\rho_f\) and \(\omega_f\). One of the main goals is to study the question: How good can an embedding be from \(X\) to \(Y\) be in the case where \(X\) and \(Y\) are some of the spaces \(L_p\), \(\ell_p\)? The author considers both the Banach space case \(p\geq 1\) and the quasi-Banach space case \(0<p<1\).
At the end of the paper the author provides a table which summarizes the known results in this direction.
The author uses related techniques for studying embeddability of locally compact amenable groups and spaces with Yu’s property A.