Suppose that $(X,d\_X)$ and $(Y,d\_Y)$ are complete doubling metric spaces and let $f:X\rightarrow Y$ be a non-constant Lipschitz map. Then can does there exist a lsc function
$\rho:(0,\infty)\rightarrow (0,\infty)$ with the properties that


* $
\lim\limits\_{t \uparrow \infty}\rho(t) = \infty
$
* $
\lim\limits\_{t \downarrow 0}\rho(t) = 0,
$


Such that for every $x,y \in X$,
$$
\rho(d\_X(x,y)) \leq d\_Y(f(x),f(y)) \leq Lip(f) d\_X(x,y)
,
$$
where $Lip(f)$ is the best Lipschitz constant of $f$. 



Lower Estimate of A Lipschitz Map
Of course, there might be such a function $\rho$ for a specific $f$, but there need not be one in general.


Let $X$ be $\mathbb{R}^2$, $Y$ be $\mathbb{R}$ and let $f(x,y)$ be $x$ if $x<0$ and $0$ if $x\geq 0$. Then, if I understood correctly, such a $\rho$ cannot exist. After all, $f$ identifies pairs of points with arbitrary distances.


