Several Complex Variables Problem List — Problem 26

Definitions and mathematical statement

The supplied source asks: "Solve $\bar\partial$--Neumann in $L^2(d\mu)$, $d\mu$ a measure, even on the unit disc." Here the unit disc is \[ \mathbb D=\{z\in{\mathbb{C}}:|z|<1\}. \] For a positive measure $\mu$ on a specified measurable space, $L^2(\mu)$ denotes the space of equivalence classes of measurable functions $f$ satisfying $\int |f|^2\,d\mu<\infty$. Positivity and the underlying measurable space are necessary specifications for this usual Hilbert-space interpretation; the source does not explicitly provide them.

A conditional operator formulation on $\mathbb D$ can be described as follows. Suppose suitable spaces $H_0=L^2(\mathbb D,\mu)$ and $H_1=L^2_{(0,1)}(\mathbb D,\mu)$ have been specified, together with a densely defined closed realization \[ T:\operatorname{Dom}(T)\subset H_0\longrightarrow H_1 \] of $\bar\partial$. Let $T^*$ be its Hilbert-space adjoint. In complex dimension one, the corresponding operator on $(0,1)$-forms is \[ \Box_{\mu,1}=TT^*,\qquad \operatorname{Dom}(\Box_{\mu,1}) =\{u\in\operatorname{Dom}(T^*):T^*u\in\operatorname{Dom}(T)\}. \] One possible solvability target is: for every \[ f\in H_1\cap(\ker\Box_{\mu,1})^\perp, \] find $u\in\operatorname{Dom}(\Box_{\mu,1})\cap(\ker\Box_{\mu,1})^\perp$ such that $\Box_{\mu,1}u=f$. Whether a bounded solution operator or further estimates are required is unspecified.

Unresolved scope. This conditional formulation illustrates a possible meaning of the source request; the source does not select the form degree, operator realization, boundary conditions, admissible measures, or meaning of "solve." For an arbitrary measure, even a distributional realization of $\bar\partial$ on $L^2(\mu)$ equivalence classes is not automatic. Consequently the supplied record does not determine a single precise quantified conjecture. The phrase "even on the unit disc" identifies a special case of interest but leaves the broader domain class unspecified.

Short English statement

The source asks for an $L^2(d\mu)$ theory of the $\bar\partial$--Neumann problem for a measure $\mu$, emphasizing the unit disc as an already interesting case. The admissible measures, operator domains, boundary conditions, form degree, and desired solvability or estimates are unspecified. The displayed operator formulation is conditional and requires human review.

Sources

• Local UnsolvedMath registry, permanent ID 800229, exact source problem code catalogue entry; title: Several Complex Variables Problem List --- Problem 26. The supplied record lists Buzzard, Fornaess, and Krantz under proposed_by, with proposed year 1995.
• Buzzard, Fornaess, Krantz, The Several Complex Variables Problem List (1995), Problem 26. Supplied source URL: https://arxiv.org/abs/math/9509202. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not independently verified here.
• The local record labels this problem open. This supplied status is not an independently verified assessment of the literature. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.