Differentiability of the function defiend by the integral and $W^{1,1}$ space

Question

Under what conditions on $f$ the function $g(r)=\int_{B_{r}(x)}u(y)dy$ is differentiable with respect to $r.$ Moreover, under what conditions on $f,$ the function $g\in W^{1,1}(0,2),$ where $W^{1,1}$ is the soblev space.

Answer 1

Hint: $\frac{d}{dr} \big( \int_{B_r(x)} f\,dx\Big) = \int_{\partial B_r(x)} f dS$, whenever $f$ is continuous and integrable.