Proposition 5.27.23.5B874925BCA1 Let $G$ be the zero extension of $\chi ''(\log |f|)|\nabla \log |f||^2$ from $U\setminus Z$, and put $\ell =\lim _{s\to -\infty }\chi '(s)$. For each $1\leq p<\infty $,
\[ G\in L^ p_{\mathrm{loc}}(U) \quad \Longleftrightarrow \quad \int _{-\infty }^ A|\chi ''(s)|^ p \mathrm e^{2(1-p)s/m_0}\, \mathrm ds<\infty . \]
In the sense of distributions,
\[ \Delta v=G+\ell \, \Delta \log |f|, \]
where $\Delta \log |f|$ is a positive measure carried by the divisor of $f$. In particular, $\Delta v$ is represented by a locally $L^ p$ function exactly when $\ell =0$ and the integral above is finite.
Proof.
The limit $\ell \geq 0$ exists because $\chi '$ is nonnegative and nondecreasing. Near a smooth point of a component of multiplicity $m_0$, outside the other components, the coordinate reduction to $f=w_1^{m_0}$ makes the positive complex Hessian of $v$ off $Z$ a single factor in the normal direction. Its Euclidean traces in the two coordinate systems are uniformly comparable, so polar integration gives necessity. In general, $G=\chi ''(\log |f|)|\partial f|^2/|f|^2$ off $Z$. The coarea formula and lemma 5.27.21, now bounding the remaining $2p-2$ powers of $|\partial f|$ by the first inequality of the lemma, give
\[ \int _{K\cap \{ 0<|f|<r_ K\} }G^ p\, \mathrm dV \leq C\int _0^{r_ K}|\chi ''(\log r)|^ p r^{-1-2(p-1)/m_0}\, \mathrm dr. \]
The substitution $s=\log r$ proves sufficiency. For $p=1$ the integral is always finite, since $\int _{-\infty }^ A\chi ''=\chi '(A)-\ell $.
For the distributional identity, use $s_\varepsilon $ and $v_\varepsilon $ from the preceding proof. Direct differentiation gives
\[ \begin{aligned} \Delta v_\varepsilon & =\chi ''(s_\varepsilon ) \frac{|f|^2|\partial f|^2}{(|f|^2+\varepsilon ^2)^2} +\chi '(s_\varepsilon )\Delta s_\varepsilon ,\\ \Delta s_\varepsilon & =\frac{2\varepsilon ^2|\partial f|^2}{(|f|^2+\varepsilon ^2)^2}. \end{aligned} \]
Off $Z$, the first term tends locally uniformly to $G$. By the level estimate of lemma 5.27.21 and the same substitution, its integral over $K\cap \{ |f|<r_0\} $ is at most
\[ C\int _{-\infty }^{\frac12\log (r_0^2+\varepsilon ^2)} \chi ''(s)\, \mathrm ds. \]
This bound tends to zero if we first let $\varepsilon \to 0$ and then $r_0\to 0$, so the first term converges to $G$ in $L^1_{\mathrm{loc}}$.
The positive measures $\Delta s_\varepsilon $ converge weakly to $\Delta \log |f|$, with locally bounded masses, and the Poincaré–Lelong formula shows that this limit is a measure supported on $Z$. On $\{ |f|<r_0\} $,
\[ 0\leq \chi '(s_\varepsilon )-\ell \leq \chi '(\tfrac 12\log (r_0^2+\varepsilon ^2))-\ell , \]
while $\Delta s_\varepsilon \to 0$ uniformly on compact subsets of $\{ |f|\geq r_0\} $. Taking the same successive limits shows that $(\chi '(s_\varepsilon )-\ell )\Delta s_\varepsilon \to 0$ weakly. Since $v_\varepsilon \to v$ in $L^1_{\mathrm{loc}}$, passing to the limit in the formula for $\Delta v_\varepsilon $ gives the identity. If $\ell >0$, the measure $\ell \, \Delta \log |f|$ is nonzero and singular with respect to Lebesgue measure; uniqueness of the Lebesgue decomposition therefore proves the final criterion.
$\square$
Pome
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