On a class of generalized Cauchy–Riemann systems

Z. A. Arushanyan

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 15-20

Résumé

The article deals with the fbllowing generalized Cauchy–Riemann equation \begin{gather} A\frac{\partial u}{\partial x}+B\frac{\partial u}{\partial y}+C\frac{\partial u}{\partial z}=0, \end{gather} where $A, B, C$ are constant $(k\times k)$ matrices such that the system (1) has only harmonic ($\mathbb R^k$-valued) solutions. For such harmonic functions $u$ the Hardy class $H^p(\mathbb R^3_+)$ is defined. A connection of this class with the Hardy class $H^1(\mathbb R^2)$ defined by Е. Stein and G. Weiss is descussed. There is obtained the following analog of the W. Rudin theorem: every compact set $E\subset\mathbb R^2$ of zero measure is an interpolation set for the space $C(\bar{\mathbb R}^3)\cap H^1(\mathbb R^3_+)$.