Corollary 23.1.55`0ACED0B738A9`

Corollary 23.1.55.0ACED0B738A9 Let $W:[0,\infty )\to (0,\infty )$ be nonincreasing. A uniform bound for the weighted masses

\[ \int _ X W(-u)\mathrm{MA}^{\mathrm{ext}}_\omega (u) \]

over all normalized $\omega $-psh potentials requires

\[ \sup _{T\gg 1}\sqrt T\, W(T)<\infty . \]

Uniform deep tails over that class require $\sqrt T\, W(T)\to 0$.

Proof. On $K_\eta $, monotonicity of $W$ gives

\[ \int _ XW(-u_\eta )\mathrm{MA}^{\mathrm{ext}}_\omega (u_\eta ) \geq D\eta ^{-1}W(B\eta ^{-2}). \]

Every sufficiently large $T$ equals $B\eta ^{-2}$ for some sufficiently small $\eta $, proving the necessary bound. Also $K_\eta \subset \{ -u_\eta >A\eta ^{-2}/2\} $, so uniform deep tails force this lower bound to tend to zero. $\square$

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BibTeX citation

@misc{pome-tag-0ACED0B738A9,
  author = {{Pome contributors}},
  title = {Corollary 23.1.55},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/0ACED0B738A9/}
}

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Corollary 23.1.550ACED0B738A9

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Corollary 23.1.55`0ACED0B738A9`