Corollary 23.1.63.E636296BB209 For a weight $W$ satisfying the assumptions of this section and agreeing for large $t$ with
\[ W_{p,a}(t)=(1+t)^{-p}[\log (\mathrm e+t)]^{-a}, \]
universal individual finiteness and uniform total-mass bounds in $\mathcal R$ hold exactly when $p>1/2$, or $p=1/2$ and $a\geq 0$. Uniform deep tails, uniform capacity absolute continuity and capacity-to-weak continuity hold exactly when $p>1/2$, or $p=1/2$ and $a>0$.
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