Several Complex Variables Problem List — Problem 18

Definitions and mathematical statement

Let $\mathbb H$ be the real quaternion skew field, with quaternionic conjugation $a\mapsto\bar a$, and let $\Omega\subset{\mathbb{R}}^4={\mathbb{C}}^2$ be a domain, meaning a nonempty connected open set. Write its real coordinates as $x_0,x_1,x_2,x_3$. Let $\Phi=(\Phi^0,\Phi^1,\Phi^2,\Phi^3)$ be a family of quaternions satisfying \[ \Phi^p\overline{\Phi^q}+\Phi^q\overline{\Phi^p}=2\delta_{p,q}\qquad(0\le p,q\le3), \] where $\delta_{p,q}$ is the Kronecker delta. For $f\in C^1(\Omega;\mathbb H)$, define \[ D_\Phi f=\sum_{k=0}^3\Phi^k\frac{\partial f}{\partial x_k},\qquad M_\Phi(\Omega;\mathbb H)=\{f\in C^1(\Omega;\mathbb H):D_\Phi f=0\}. \] The coefficients multiply the derivatives on the left; members of this kernel are called left-$\Phi$-hyper-holomorphic functions.

Let $i,j,k$ be the standard quaternionic units, satisfying $i^2=j^2=k^2=ijk=-1$, and put $\hat\Phi=(1,i,j,-k)$. The source asserts \[ \operatorname{Hol}(\Omega;{\mathbb{C}}^2)\subset M_{\hat\Phi}(\Omega;\mathbb H). \] This comparison requires an identification of the target ${\mathbb{C}}^2$ with $\mathbb H$. To make the notation explicit without selecting an unstated convention, denote that identification by a fixed real-linear bijection $\iota:{\mathbb{C}}^2\to\mathbb H$, and write \[ \mathcal H_\iota(\Omega)=\{\iota\circ F:F\in\operatorname{Hol}(\Omega;{\mathbb{C}}^2)\}. \] Here holomorphicity is taken with respect to the source's identification ${\mathbb{R}}^4={\mathbb{C}}^2$. The source does not explicitly specify these domain and target conventions; its asserted inclusion is understood relative to compatible conventions, not as an assertion for every $\iota$.

The requested descriptions are as follows.

1. Describe the hyper-holomorphic functions that are not holomorphic. Under fixed identification conventions, the relevant difference is \[ \mathcal N_{\Phi,\iota}(\Omega)=M_\Phi(\Omega;\mathbb H)\setminus\mathcal H_\iota(\Omega). \] The source says that the requested set is nonempty. Its immediately preceding inclusion concerns $\hat\Phi$, but it does not consistently specify whether this question concerns only $\hat\Phi$ or arbitrary admissible $\Phi$.
2. Describe the "locations" of the sets denoted in the source by $Hol$ and $M_\Phi Hol$ inside $M_\Phi$. The expression $M_\Phi Hol$ is not defined in the supplied record, and "locations" is not assigned a topology, algebraic structure, or other precise meaning. Moreover, the supplied inclusion places holomorphic maps inside the kernel only for $\hat\Phi$, not for arbitrary $\Phi$. Thus this part remains an unresolved formulation request; in particular, $M_\Phi Hol$ is not silently interpreted as a set difference or a product.
3. Describe how the sets and their relationships in the preceding questions depend on the domain $\Omega$. The source specifies no particular class of domains or equivalence relation for comparing different domains.

The identification conventions, the intended range of $\Phi$, and the meaning of the second request require clarification before all three requests constitute a uniquely specified mathematical problem.

Short English statement

Describe quaternion-valued solutions of the left differential equation $D_\Phi f=0$ that do not arise from ordinary holomorphic maps, their relationship to holomorphic maps, and their dependence on $\Omega$. The source singles out $\hat\Phi=(1,i,j,-k)$, but leaves identification conventions and part of the requested comparison unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $800222$, source problem code catalogue entry, titled Several Complex Variables Problem List --- Problem 18; collection: AMR Open Problem Lists. The registry labels the record "open"; this is supplied classification, not an independently verified assessment.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 18. Supplied source URL: https://arxiv.org/abs/math/9509202. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance notes, not a claim of independent source inspection.
• The problem text is signed by H. Shapiro and N. Vasilievsky, with the supplied contact matemat@cinvesmx.bitnet. The registry's proposed_by field names Buzzard, Fornaess, and Krantz and gives the year 1995; both attribution layers are retained.