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Given a Hörmander system $X=\{X_1,\cdots ,X_m\}$ on a domain $\Omega \subseteq {𝐑}^n$ we show that any subelliptic harmonic morphism $\phi$ from $\Omega$ into a $\nu$-dimensional riemannian manifold $N$ is a (smooth) subelliptic harmonic map (in the sense of J. Jost & C-J. Xu, [9]). Also $\phi$ is a submersion provided that $\nu \le m$ and $X$ has rank $m$. If $\Omega ={𝐇}_n$ (the Heisenberg group) and $X=\left\{\frac{1}{2}\left(L_{\alpha }+L_{\overline{\alpha }}\right),\frac{1}{2i}\left(L_{\alpha }-L_{\overline{\alpha }}\right)\right\}$, where $L_{\overline{\alpha }}=\partial /\partial {\overline{z}}^{\alpha }-i{z}^{\alpha }\partial /\partial t$ is the Lewy operator, then a smooth map $\phi :\Omega \to N$ is a subelliptic harmonic morphism if and only if $\phi \circ \pi :(C\left({𝐇}_n\right),F_{{\theta }_0})\to N$ is a harmonic morphism, where $S^1\to C\left({𝐇}_n\right)\stackrel{\pi }{\to }\to {𝐇}_n$ is the canonical circle bundle and $F_{{\theta }_0}$ is the Fefferman...

We establish an inversion formula for the M. M. Djrbashian A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain ${\Omega }_n=\zeta \in {ℂ}^n:\varrho \left(\zeta \right)>0$, $\varrho \left(\zeta \right)=Im\left({\zeta }_1\right)-{\left|{\zeta }^{\text{'}}\right|}^2$. We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the ${\varrho }^{\alpha }$-Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of ${\Omega }_n$. We build an anti-holomorphic embedding of ${\Omega }_n$ in the complex projective Hilbert space $ℂ\mathbb{P}\left(H_{\alpha }^2\left({\Omega }_n\right)\right)$ and study (in connection with work by A. Odzijewicz [18] the corresponding transition probability amplitudes....