Characterization of Sobolev regularity of plurisubharmonic functions

Abstract

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 $v=\chi(\log|f|).$ The criteria for the $L^p$-integrability of the classical Laplacian and for $W^{1,p}$-regularity are given by a weighted integral involving $\chi''$ and $\chi'$, respectively, and depend on $f$ only through the smallest multiplicity of $\operatorname{Div}(f)$. We also obtain the $W^{2,p}_{\mathrm{loc}}$ criterion for $1<p<\infty$. At the endpoint $p=1$, we prove that \[ v\in W^{2,1}_{\mathrm{loc}} \quad\Longleftrightarrow\quad \chi'\in L^1((-\infty,A)), \] equivalently, $v$ is locally bounded. As applications, we obtain counterexamples to Calder\'on-Zygmund theory and characterize a class of functions in the local Monge--Amp\`ere domain. % and exhibit a natural family in $W^{2,1}_{\mathrm{loc}}$ whose limit fails to belong to $W^{2,1}_{\mathrm{loc}}$.