Dimension of images and graphs of little Lipschitz functions
Volume 262 / 2023
Abstract
A mapping $f\colon X\to Y$ between metric spaces is termed little Lipschitz if the function ${\rm lip}\, f\colon X\to [0,\infty ]$, $${\rm lip}\, f(x)=\liminf_{r\to 0}\frac{{\rm diam}\,f(B(x,r))}{r},$$ is finite at every point. We prove that for each $s \gt 0$ the little Lipschitz mapping $f$ satisfies the inequality $$ \mathscr H^s(f(X))\leq \int _X({\rm lip}\, f)^s\,{\rm d}\mathscr P^s $$ as long as $\{{\rm lip}\, f=0\}$ is of $\sigma $-finite measure $\mathscr P^s$, where $\mathscr H^s$ and $\mathscr P^s$ denote the $s$-dimensional Hausdorff and packing measures, respectively. We derive a dimensional inequality for little Lipschitz mappings $$\dim _{\mathsf H} f(X)\leq \dim _{\mathsf H} f\leq \mathop{\overline {\rm dim}_{\mathsf P}} X$$ and we provide a few examples that show that these inequalities are the best possible.
