I've been stuck on this one for a while now.
Given an $\mathcal{H}^{s}$ measurable subset $E\subset \mathbb{R}^n$ with $0<\mathcal{H}^{s}(E)<\infty$, we let $\overline{D}^{s}(E,x)$ denote the upper spherical density of $E$, defined as $$\lim\sup_{r\rightarrow 0} \frac{\mathcal{H}^{s}(E\cap B_r(x))}{(2r)^s}$$
I'm trying to prove that given a $\mathcal{C}^1$ map $\phi$ with nonzero Jacobian, we have that $\overline{D}^{s}(E,x) = \overline{D}^{s}(\phi(E),\phi(x))$ for $\mathcal{H}^{s}$ almost all $x$.
I've been considering just linear maps so far. I tried isolating pieces of $E$ whose $\mathcal{H}^s$ measure could be approximated well using subsets $D\subset \mathbb{R}^{n}$ whose diameters changed by a factor of, say ($d\pm \varepsilon$) (for small $\varepsilon$) in an attempt to get estimates on $\mathcal{H}^s(\phi(E))$, but I've been unsuccessful so far. In the end the only estimates I've been able to obtain have been those which use the constants $M,m$ satisfying $m|x-y|\leq |\phi(x)-\phi(y)|\leq M|x-y|$, but this only takes me so far in proving the equality of the densities. I also attempted to relate the spherical density of $\phi(E)$ to the density of $E$ with respect to the family $\mathcal{A}$ consisting of those sets $A$ such that $\phi(A)$ is a ball, but for this to work I need estimates relating the latter density and $\overline{D}^{s}(E,x)$.
Any help would be greatly appreciated.
