Characterization of Sobolev regularity of plurisubharmonic functions

Let $f$ be a nonzero holomorphic germ at $0\in {\mathbb{C}}^n$ with $f(0)=0$, and let $\chi$ be a $C^2$ non-decreasing convex function on the left half-line. We establish sharp necessary and sufficient conditions for the local Sobolev regularity of the plurisubharmonic function $v=\chi (\log \mathrm{\mid }f\mathrm{\mid })\mathrm{.}$ The criteria for the $L^p$-integrability of the classical Laplacian and for $W^{1,p}$-regularity are given by a weighted integral involving ${\chi }^{\mathrm{\prime }\mathrm{\prime }}$ and ${\chi }^{\mathrm{\prime }}$, respectively, and depend on $f$ only through the smallest multiplicity of $\mathrm{Div}(f)$. We also obtain the $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,p}$ criterion for $1<p<\mathrm{\infty }$. At the endpoint $p=1$, we prove that $$v\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,1}⟺{\chi }^{\mathrm{\prime }}\in L^1((-\mathrm{\infty },A)),$$ equivalently, $v$ is locally bounded. As applications, we obtain counterexamples to Calderón-Zygmund theory and characterize a class of functions in the local Monge–Ampère domain. % and exhibit a natural family in $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,1}$ whose limit fails to belong to $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,1}$.