Ufa Mathematical Journal
    Volume 13, Number 3, pp. 191-205

    The boundary Morera theorem for the domain ${{\tau }^{+}}\left( n-1 \right)$

    G. Khudayberganov, J.Sh. Abdullayev


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    In this work, we will continue to construct an analysis in the future tube and move on to study the Lie ball. The Lie ball can be realized as a future tube. These realizations will be the subject of our research. These methods turn out to be convenient for computing the Bergman, Cauchy-Szeg\"{o} and Poisson kernels in this domain. In the theory of functions Morera's theorems have been studied by many mathematicians. In the complex plane, the functions with one-dimensional holomorphic extension property are trivial but Morera's boundary theorems are not available. Therefore, the results of the work are essential in the multidimensional case. In this article, we proved the boundary Morera theorem for the domain ${{\tau }^{+}}\left( n-1 \right)$. An analog of Morera's theorem is given, in which integration is carried out along the boundaries of analytic disks. For this purpose, we use the automorphisms ${{\tau }^{+}}\left( n-1 \right)$ and the invariant Poisson kernel in the domain ${{\tau }^{+}}\left( n-1 \right)$. Moreover, an analogue of Stout's theorem on functions with the one-dimensional holomorphic continuation property is obtained for the domain ${{\tau }^{+}}\left( n-1 \right)$. In addition, generalizations of Tumanov's theorem is obtained for a smooth function from the given class of CR manifolds.