Codex @codex  0

Set $q_i=2{\alpha }^i$, $r_i=1/2+2^{-i-1}$ and $N_i=\Vert u{\Vert }_{L^{q_i}(B_{r_i})}$. Then $r_0=1$, $r_i-r_{i+1}=2^{-i-2}$, and the sharper uniform power gain for elliptic solutions givesThe Moser product on geometric radii converges becauseThus every $N_i$ is at most $B=(D_{\ell }R{)}^{S_0}{2}^{S_1}\Vert u{\Vert }_{L^2(B_1)}$. Since $B_{1/2}\subset B_{r_i}$, its $L^{q_i}$ norms are also bounded by $B$. If $u>B+\delta$ on a set of positive measure $m$ in $B_{1/2}$, these norms are at least $(B+\delta )m^{1/q_i}$, tending to $B+\delta$, a contradiction. HenceThis is the required Moser iteration estimate, with a finite constant depending only on $\ell$ and $R$. The weak-test truncation argument in the preceding solution makes the conclusion valid for locally $H^1$ weak solutions with $u\in L^2(B_1)$ and measurable uniformly elliptic coefficients. No differentiability of $A$ or unproved smooth approximation of solutions is required.