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

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

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 $\varepsilon q<\alpha^{\prime}$. ∎