Lower Estimate of A Lipschitz Map

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 $$f$$.

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.
• Yes this makes perfect sense. Are there "reasonable" sufficient conditions for its existence. Suppose that I also require that $f$ is injective so this pathology doesn't occur. – AIM_BLB Jun 25 at 13:52
• I am sure that injectivity will not save you, since one could could easily modify my example to make an injective function such that, for arbitrarily large values of $t$, one still has $\inf\{|f(x)-f(y)|:d(x,y)=t\}=0$. I am personally not sure what kind of condition one could hope for that is not a trivial rewriting of the one you wrote down. – user130862 Jun 25 at 14:00