Suppose $u$ is a nonnegatieve (but not identically zero) integrable function defined on $B_2(0)$, ball of radius $2$ and centred at $0$ such that $$ \sup_{B\frac{1}{2}(0)}\,u\leq C\inf_{B_{\frac{1}{2}(0)}}\,u $$ for some positive constant $C$.
Then I hope the following fact is also true: For any ball $B_r(0)$ where $r<2$, one has $$ \sup_{B_r(0)}\,u\leq D\inf_{B_r(0)}\,u, $$ for some positive constant $D$ depedning on the ball $B_r(0)$.
I tried to prove in the following way: Since closure of $B_r(0)$ is compact, by compactness, we can cover it by a finite number of balls (say $m$) $B$ each of radius $\frac{1}{2}$ centred at $0$.
Moreover, observe that from the given condition $$ \sup_{B\frac{1}{2}(0)}\,u\leq C\inf_{B_{\frac{1}{2}(0)}}\,u $$ for some positive constant $C$, and since $u\geq 0(\neq 0)$, we have $u\geq C_1>0$ in $B_\frac{1}{2}(0)$ and therefore $u\geq C_2$ in $B_r(0)$ for some positive constant $C_2$.
Now since $B_r(0)\subset\overline{B_r(0)}$ and using the above finite number of coverings, we obtain $$ \sup_{B_r(0)}\,u\leq \sup_{\overline{B_r(0)}}\,u\leq C_{m}\sup_{B_{\frac{1}{2}(0)}}\,u\leq d_{m}\inf_{B_{\frac{1}{2}(0)}}\,u\leq e_{m}\inf_{B_r(0)}\,u, $$ for some positive constants $c_m, d_m$ and $e_m$, which are finite since the number of covering is finite. In this way $e_m$ depending on $B_r$.
I hope the argument is correct.
Can somebody kindly help me if you think there is some problem in the argument?
Thanking you in advance.