Characterization of Sobolev regularity of plurisubharmonic functions
Abstract
Let f be a nonzero holomorphic germ at 0 ∈ ℂn with f(0)=0, and let χ be a C2 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 Lp-integrability of the classical Laplacian and for W1,p-regularity are given by a weighted integral involving χ'' and χ', respectively, and depend on f only through the smallest multiplicity of Div(f). We also obtain the W2,ploc criterion for 1<p<∞. At the endpoint p=1, we prove that v∈ W2,1loc Longleftrightarrow χ'∈ L1((−∞,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 W2,1loc whose limit fails to belong to W2,1loc.
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BibTeX
@misc{chen2026characterization,
title = {{Characterization of Sobolev regularity of plurisubharmonic functions}},
author = {Hongrong Chen and Guokuan Shao and Wenxuan Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39136},
eprint = {2609.39136},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39136},
}
RIS
TY - UNPB
TI - Characterization of Sobolev regularity of plurisubharmonic functions
AU - Hongrong Chen
AU - Guokuan Shao
AU - Wenxuan Wang
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39136
UR - https://arxiv.org/abs/2609.39136
AB - Let f be a nonzero holomorphic germ at 0 ∈ 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 Div(f). We also obtain the W^2,p_loc criterion for 1<p<∞. At the endpoint p=1, we prove that v∈ W^2,1_loc Longleftrightarrow χ'∈ L^1((−∞,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_loc whose limit fails to belong to W^2,1_loc.
ER -