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Corollary 23.1.63E636296BB209

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 $a>0$.

Proof. Apply theorem 23.1.62 to $\sqrt t\, W_{p,a}(t)\sim t^{1/2-p}(\log t)^{-a}$. A smooth decreasing modification on a compact interval changes none of these conditions. $\square$

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