Generalization of a theorem of M. Jarnicki about cells of harmonicity to \(\mathbb C^n\) (Q1772531)

It is well known [\textit{P. Lelong}, Bull. Soc. Math. Belg. 7, 10--23 (1954; Zbl 0068.08301)] that for every domain \(D\subset\mathbb C^n\simeq\mathbb R^{2n}\simeq \mathbb R^{2n}+i0\subset\mathbb R^{2n}+i\mathbb R^{2n}\simeq\mathbb C^{2n}\) there exists the maximal domain \(\widetilde D\subset\mathbb C^{2n}\) such that \(\widetilde D\cap\mathbb R^{2n}=D\) and every function \(h\) harmonic in \(D\) extends to a function \(\widetilde h\) holomorphic in \(\widetilde D\); the domain \(\widetilde D\) is called the harmonic envelope of holomorphy of \(D\). Let \(f=(f_1,\dots,f_{2n}):D\to D'\) be a diffeomorphism with harmonic components. Put \(\widetilde f:=(\widetilde f_1,\dots,\widetilde f_{2n}): \widetilde D\to\mathbb C^{2n}\). The author discusses some elementary cases in which \(\widetilde f:\widetilde D\to\widetilde{D'}\) is biholomorphic.