Cauchy–Fueter operators

Definition

Write $q=t+xi+yj+zk\in\mathbb H$ . For a smooth quaternion-valued function $F$ , the Cauchy–Fueter operators in the convention used here are

\frac{\partial F}{\partial\bar q} =\partial_tF+i\partial_xF+j\partial_yF+k\partial_zF, \qquad \frac{\partial F}{\partial q} =\partial_tF-\partial_xF\,i-\partial_yF\,j-\partial_zF\,k.

In several quaternionic variables one applies these formulas to each coordinate.

Relation to the Laplacian

On real-valued functions, the appropriate compositions recover the Euclidean Laplacian on $\mathbb R^4$ , up to the normalization built into the displayed operators. Mixed derivatives form the quaternionic Hessian.

Convention warning

Because $\mathbb H$ is noncommutative, left and right placement of the units $i,j,k$ matters. Authors also insert factors such as $1/2$ . A formula using $\partial_q$ or $\partial_{\bar q}$ must therefore state its convention; the two displayed operators are not obtained by treating quaternionic coefficients as if they commute.

References

Semyon Alesker, “Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables,” Bulletin des Sciences Mathématiques 127 (2003), 1–35. arXiv record. Relevant: §2.
Semyon Alesker, “Quaternionic plurisubharmonic functions and their applications to convexity,” 2016 revision. arXiv record. Relevant: §2.