i have this exercise: For all $ B_r(x)$ $\subset$$\subset$ $\Omega$ , $ u(x) \leq S(u; x, r) $ if and only if $ u(x) \leq A(u; x, r)$
when $ A(u; x, r) = \frac{n}{\omega_n r^n} \int_{B_r(x)} u(y) \, dy $ and its spherical average by $ S(u; x, r) = \frac{1}{\omega_n r^{n-1}} \int_{\partial B_r(x)} u(\sigma) \, d\sigma $
$\Rightarrow$ if $u(x) \leq S(u; x, r)$, then $\int_{B_r(x)} u(\sigma) \, d\sigma=\int_0^R \int_{ \partial B_r(x)} u(\sigma) \, d\sigma dr \geq \int_{0}^Rr^{n-1}\omega_{n}u(x)=u(x)r^{n}\frac{\omega_{n}}{n}$
so $u(x) \leq A(u,x,r)$
but for the $\Leftarrow$ I can't use a similar idea
