Lemma 15.35.8C9E4D8D9901A

Proof. In the notation of the proof of lemma 15.30.1, $t=r/(1+r)$ and $s(r)=L$. Since the mass of $q_\alpha \kappa ^ k$ on $\{ r_1<r<r_2\} $ is $Q(r_2)^ k-Q(r_1)^ k$, this measure has radial distribution $-\mathrm{d}(Q^ k)/\mathrm{d}L=W(L)\, \mathrm{d}L$. The Fubini–Study measure $\kappa ^ k$ corresponds to $Q=t$, so its radial distribution is $k\mathrm e^{-k(L-1)}\mathrm{d}L$. Both measures are invariant under the unitary group of the chart, so $q_\alpha $ is the ratio of the two distributions, which is the stated formula. The lower bound $q_\alpha \geq b_0$ follows from $\kappa +\mathrm{dd}^{\mathrm{c}}u_\alpha \geq (1-\alpha )\kappa $ off $o$. As $L\to +\infty $, $t$ tends to zero exponentially, so $Q\sim \alpha L^{\alpha -1}$ and $D\sim \alpha (1-\alpha )L^{\alpha -2}$; this gives the asymptotics of $W$. Consequently,

and the mass of $B_\tau \kappa ^ k$ on $\{ L>T\} $, which equals $\int _ T^\infty \mathrm e^{\tau (L^\alpha -1)}W(L)\, \mathrm{d}L$, is infinite for every $T$. $\square$