Failure of bounded ∂̅-solvability on pseudoconvex domains of finite type
Abstract
We show that for every integer n≥3 there is a bounded pseudoconvex domain in ℂn with smooth real-analytic boundary whose maximum boundary D'Angelo $1$-type is $16$, carrying a ∂̅-closed form continuous on the closure and smooth in the domain with no bounded distributional solution. On each fixed domain, we also construct a uniformly bounded sequence of smooth ∂̅-closed data for which every bounded solution has norm at least ck1/320 for all sufficiently large k. Each datum in this sequence and a corresponding solution extend smoothly to a neighborhood of the closure. The obstruction comes from a singular curve in two weak complex tangential directions, whose normalization admits holomorphic functions that do not extend to a neighborhood of the curve.
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BibTeX
@misc{du2026failure,
title = {{Failure of bounded $\bar\partial$-solvability on pseudoconvex domains of finite type}},
author = {Yun-Heng Du},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03127},
eprint = {2611.03127},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03127},
}
RIS
TY - UNPB
TI - Failure of bounded ∂̅-solvability on pseudoconvex domains of finite type
AU - Yun-Heng Du
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03127
UR - https://arxiv.org/abs/2611.03127
AB - We show that for every integer n≥3 there is a bounded pseudoconvex domain in ℂ^n with smooth real-analytic boundary whose maximum boundary D'Angelo $1$-type is $16$, carrying a ∂̅-closed form continuous on the closure and smooth in the domain with no bounded distributional solution. On each fixed domain, we also construct a uniformly bounded sequence of smooth ∂̅-closed data for which every bounded solution has norm at least ck^1/320 for all sufficiently large k. Each datum in this sequence and a corresponding solution extend smoothly to a neighborhood of the closure. The obstruction comes from a singular curve in two weak complex tangential directions, whose normalization admits holomorphic functions that do not extend to a neighborhood of the curve.
ER -