Abstract:

Let $D$ be a bounded domain in $\mathbb C^n\ (n>1)$ with a connected infinitely smooth boundary $\Gamma$, and the function $f$ is harmonic in $D$ and of class $\mathcal C^1(\bar D)$. For a vector field $w$ (not lying in the complex tangent plane to $\Gamma$) the differential condition $\bar w(f)=\sum\limits_{k=1}^n\bar w_k\dfrac{\partial f}{\partial\bar z_k}=0$ by $\Gamma$ is considered. Will $f$ be holomorphic in $D$? This problem is an analogue of the problem with an oblique derivative for real-valued harmonic functions. The paper shows that this problem is connected with a certain Cauchy-Fantappié integral representation $Q$, the kernel of which consists of derivatives of the fundamental solution of the Laplace equation. Under some additional conditions on the vector field $w$, it is shown that the iterations of $Q^m$ of this Cauchy-Fantappié integral representation converge to a holomorphic function. By doing so the problem under consideration has a positive reply.