It is enough to establish ${\mathcal{H}}(\alpha,A_{\alpha},\omega)$ for compact subsets, by regularity of $\mu$ and $Cap_{\omega}$. Let $K$ be a compact subset of $X$. It follows from Hölder’s inequality that

$$0\leq\mu(K)\leq||f||_{L^{p}(\omega^{n})}\left[\text{Vol}_{\omega}(K)\right]^{1/q},$$

where $1/p+1/q=1$. Note that $||f||_{L^{p}(\omega^{n})}=1$ since we assume $\mu$ is a probability measure. We claim that

(4)

$$\text{Vol}_{\omega}(K)\leq C_{\omega}\exp\left[-(Cap_{\omega}(K))^{-\gamma_{\omega}/n}\right],$$

for some constants $C_{\omega},\gamma_{\omega}>0$ that only depend on $\omega$. We will be done if we can prove (4) since we can then check by elementary computations that $\exp(-x^{-\delta})$ is dominated from above by $A_{\alpha}x^{\alpha}$, for all $x\in[0,1]$.

The set of functions ${\mathcal{F}}_{0}:=\{\varphi\in PSH(X,\omega)\,/\,\sup_{X}\varphi=0\}$ is compact in $L^{1}(X)$ (see proposition 2.7, [GZ 1]). These functions have Lelong numbers $\nu(\varphi,x)\leq\nu_{\omega}$ bounded from above by a uniform constant. It follows therefore from Skoda’s uniform integrability theorem [Z], that

$$\sup_{\varphi\in{\mathcal{F}}_{0}}\int\exp\left[-\frac{2\varphi}{\nu_{\omega}+1}\right]\omega^{n}\leq C_{2}<+\infty.$$

Set $\gamma_{\omega}:=2/(\nu_{\omega}+1)>0$ and let

$$V_{K,\omega}^{*}(x):=\left(\sup\{\varphi(x)\,/\,\varphi\in PSH(X,\omega),\,\varphi\leq 0\text{ on }K\}\right)^{*}$$

denote the Siciak extremal function of $K$ (see section 5.1 in [GZ 1]). Then

$$Vol_{\omega}(K)\leq\int_{X}\exp\left(-\gamma_{\omega}V_{K,\omega}^{*}\right)\omega^{n}\leq C_{2}T_{\omega}(K)^{\gamma_{\omega}},$$

where $T_{\omega}(K):=\exp(-\sup_{X}V_{K,\omega}^{*})$ denote the Alexander capacity of $K$ (see section 5.2 in [GZ 2]). It follows now from theorem 7.1 in [GZ 1] that

$$T_{\omega}(K)\leq e\exp\left[-Cap_{\omega}(K)^{-1/n}\right],$$

which yields (4). ∎

Observe that $\omega^{n}=Ed\lambda$ for some smooth density $E\geq 0$ which vanishes along the exceptional divisor of $\pi$, thus

$$\mu=f_{1}d\lambda=f\omega^{n},\text{ where }f=f_{1}/E.$$

Fix local coordinates $(z^{i})_{1\leq i\leq n}$ on a polydisk $\mathbb{D}\subset X$ and a local embedding $F:V\to\mathbb{C}^{m}$. Note that $E$ is comparable to $|\frac{\partial F}{\partial z^{1}}\wedge\ldots\wedge\frac{\partial F}{\partial z^{1}}|^{2}\simeq\sum_{i=1}^{r}|f_{i}|^{2}$, $f_{i}$ being holomorphic on $\mathbb{D}$. Therefore $E\in L^{\infty}_{loc}(\mathbb{D})$ and $E^{-\alpha}\in L_{loc}^{1}(\mathbb{D},d\lambda)$ for some $0<\alpha<1$.

Choose $0<\alpha^{\prime}<\alpha$ such that $\frac{1}{p}+\frac{1}{\alpha}=\frac{1}{\alpha^{\prime}}$. Then $f^{\alpha^{\prime}}=f_{1}^{\alpha^{\prime}}E^{-\alpha^{\prime}}$ is the product of a function in $L^{p/\alpha^{\prime}}(d\lambda)$ and a function in $L^{\alpha/\alpha^{\prime}}(d\lambda)$, hence it is in $L^{1}_{loc}(\mathbb{D},d\lambda)$ by Hölder’s inequality. A second application of Hölder’s inequality yields

$$\int_{{\mathbb{D}}}f^{1+\varepsilon}\omega^{n}=\int_{\mathbb{D}}f^{\epsilon}f_{1}d\lambda\leq\left(\int_{\mathbb{D}}f^{\epsilon q}d\lambda\right)^{1/q}\left(\int_{\mathbb{D}}f_{1}^{p}d\lambda\right)^{1/p}<+\infty,$$

where $q$ denotes the conjugate exponent to $p$. This shows that $f\in L^{p^{\prime}}(\omega^{n})$ if $p^{\prime}=1+\varepsilon>1$ is chosen so small that $\varepsilon q<\alpha^{\prime}$. ∎