= Uniform power gain for elliptic solutions
{title2=$\|u\|_{q\alpha,r_1}\le[D_\ell R/(r_2-r_1)]^{2/q}\|u\|_{q,r_2}$}
For a nonnegative $H^1$ <weak solution> of a <homogeneous divergence-form elliptic equation> on $B_1\subset\mathbb R^\ell$, $\ell\ge2$, with symmetric measurable $\lambda I\le A\le\Lambda I$, set $R=\Lambda/\lambda$ and $\alpha=1+1/\ell$. There is $D_\ell\ge1$ independent of $q\ge2$ such that
$$
\|u\|_{L^{q\alpha}(B_{r_1})}\le\left(\frac{D_\ell R}{r_2-r_1}\right)^{2/q}\|u\|_{L^q(B_{r_2})},\qquad0<r_1<r_2\le1.
$$
Test the equation with a cutoff times $u^{q-1}$, first regularized or truncated, to get $\int\zeta^2|\nabla u^{q/2}|^2\le[q/(q-1)]^2R^2\int u^q|\nabla\zeta|^2$. Apply a fixed-domain <Sobolev embedding theorem> to $\zeta u^{q/2}$ and take the $2/q$ power. Keeping the constant inside that power is crucial for <Moser iteration>.
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