= Moser product on geometric radii
{c}
{title2=$\prod_{i\ge0}[DR2^{i+2}]^{\alpha^{-i}}$}
For $q_i=2\alpha^i$, $r_i=1/2+2^{-i-1}$, a <uniform power gain for elliptic solutions> bounds the successive norms by factors $[DR2^{i+2}]^{\alpha^{-i}}$. Their infinite product is
$$
(DR)^{\alpha/(\alpha-1)}2^{\alpha/(\alpha-1)^2+2\alpha/(\alpha-1)}.
$$
It is finite because both $\sum\alpha^{-i}$ and $\sum i\alpha^{-i}$ converge. Uniform control of the $L^{q_i}$ norms on $B_{1/2}$ then controls the <essential supremum>, by comparing with the measure of a set where the function exceeds the proposed bound. A constant independent of $q$ left outside the $2/q$ power at every step would instead give a divergent product.
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