Scaling of Harnack inequality

Question

It is well known that if $u$ is a positive harmonic function in $B_R$, there exists $C>0$ such that $$ \sup_{B_{R/2}} u \leq C \inf_{B_{R/2}} u. $$ Moreover, for every $\delta \in (0,1)$, one can slightly refined the previous result by showing $$ \sup_{B_{(1-\delta)R}} u \leq \frac{C}{\delta^n} \inf_{B_{(1-\delta)R}} u. $$ This estimate can be easily computed in the case of harmonic functions by using the representation formula of $u$ in $B_R$. Does exist a similar inequality (the case $\delta \in (0,1)$) for the case of solution of a linear non-divergence form uniformly elliptic equation $$ \sum_{ij}a_{ij}(x) D_{ij} u =0 \quad\mbox{in }B_R? $$ Moreover, in the case of non-negative supersolution of this last equation, one can prove the so called weak Harnack inequality, i.e. there exists $C$ such that $$ \left(\frac{1}{|B_{R/2}|}\int_{B_{R/2}}u^\varepsilon \right)^{1/\varepsilon} \leq C \inf_{B_{R/2}}u. $$ In this setting, is it possibile to have a similar estimate in $B_{(1-\delta)R}$ with the explicit constant?