发表机构
School of Mathematics (Zhuhai), Sun Yat-Sen University; Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中山大学数学学院（珠海）; 中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文刻画了由全纯芽构造的多次调和函数的局部Sobolev正则性，给出尖锐判据，并应用于Calderón-Zygmund理论反例与Monge-Ampère定义域刻画。
AI 中文摘要
设$f$为$0 \in \mathbb C^n$处的非零全纯芽且$f(0)=0$，并设$\chi$为左半直线上$C^2$非降凸函数。我们建立了多次调和函数$v=\chi(\log|f|)$的局部Sobolev正则性的尖锐充分必要条件。经典Laplacian的$L^p$可积性和$W^{1,p}$正则性的判据分别由涉及$\chi''$和$\chi'$的加权积分给出，且仅通过$\operatorname{Div}(f)$的最小重数依赖于$f$。我们还获得了$1<p<\infty$时的$W^{2,p}_{\mathrm{loc}}$判据。在端点$p=1$处，我们证明\\[ v\in W^{2,1}_{\mathrm{loc}} \quad\Longleftrightarrow\quad \chi'\in L^1((-\infty,A)), \\] 等价地，$v$是局部有界的。作为应用，我们获得了Calderón-Zygmund理论的反例，并刻画了局部Monge-Ampère定义域中的一类函数。
英文摘要
Let $f$ be a nonzero holomorphic germ at $0 \in \mathbb C^n$ with $f(0)=0$, and let $χ$ 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 $v=χ(\log|f|).$ The criteria for the $L^p$-integrability of the classical Laplacian and for $W^{1,p}$-regularity are given by a weighted integral involving $χ''$ and $χ'$, 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 \[ v\in W^{2,1}_{\mathrm{loc}} \quad\Longleftrightarrow\quad χ'\in L^1((-\infty,A)), \] 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 $W^{2,1}_{\mathrm{loc}}$ whose limit fails to belong to $W^{2,1}_{\mathrm{loc}}$.
