## Karakalpak Scientific Journal

#### Abstract

In the theory of functions of one complex variable, Morer's theorem is known, which is inverse in some sense to the classical Cauchy theorem. On the complex plane, results on functions with the one-dimensional property of holomorphic continuation are trivial, and Morer's boundary theorems are absent. We note that the ordinary (non-boundary) Morera theorems in domains of space are well known. The first result related to our topic was obtained by Agranovsky M.L. and Valsky R.E. [1], who studied functions with the one-dimensional property of holomorphic continuation in a ball. The proof was based on the properties of the automorphism group of a ball. By Stout E.L., who used the complex Radon transform, the Agranovsky and Walski theorem was carried over to arbitrary bounded domains with a smooth boundary [4]. An alternative proof of Stout's theorem was obtained by Kytmanov A.M. [6], who applied the Bochner – Martinelli integral. The idea of using integral representations (Bochner – Martinelli, Cauchy – Fantappier, logarithmic residue) proved to be useful in studying functions with the one-dimensional property of holomorphic continuation along complex curves. A weaker property than the property of one-dimensional holomorphic continuation is the so-called Morera property. It consists in the vanishing of the integrals of a given function over the intersection of the boundary of the region with complex lines (complex planes). Greenberg E. [3] studied functions with the Morera property in a ball. Globevnik I. and Stout E.L. [4] obtained Morer's boundary theorem for an arbitrary bounded domain. In this paper, we consider the Morera boundary theorem for one Siegel domain of the second kind defined in the space of complex rectangular matrices. The proof is based on the properties of the Poisson integral for the Siegel domain, and the Cayley transform is also used.

#### First Page

25

#### Last Page

34

#### References

1. Agranovsky M.L., Valsky R.E. Maximality of invariant function algebras. Sibirsk. Mat. Zh., 1971, vol. 12, no. 1, pp. 3-12. 2. Nagel A., Rudin W. Moebius-invariant function spaces on balls and spheres. Duke Math. J., 1976, vol. 43, no. 4, pp. 841-865. 3. Grinberg E. A boundary analogue of Morera's theorem on the unit ball of . Proc. Amer. Math. Soc., 1988, vol. 102, pp. 114-116. 4. Globevnik J., Stout E. L. Boundary Morera theorems for holomorphic functions of several complex variables. Duke Math. J., 1991, vol. 64, no. 3, pp. 571-615. 5. Globevnik J. A boundary Morera theorem. J. Geom. Anal., 1993, vol. 3, no. 3, pp. 269-277. 6. Kytmanov A. M., Myslivets S. G. On a boundary analogue of the Morera theorem. Sibirsk. Mat. Zh., 1995, vol. 36, no. 6, pp. 1350-1353. 7. Kosbergenov S., Kytmanov A. M., Myslivets S. G. On the Morera boundary theorem for classical domains. Sibirsk. Mat. Zh., 1999, vol. 40, no. 3, pp. 595-604. 8. Khudayberganov G., Kytmanov A. M., Shaimkulov B. A. Complex analysis in the matrix domains. Krasnoyarsk: Siberian Federal University, 2011. 9. Hua Lo-ken. Harmonic analysis of functions of several complex variables in classical domains. Moscow, IL, 1959. 10. Khudayberganov G., Kurbanov B. T. A realization of the classical domain of the first type. Uzbek. Mat. Zh., 2014, no. 1, pp. 126-129. 11. Koranyi A. The Poisson integral for the generalized half planes and bounded symmetric domains. Ann. of Math. (2)., 1965, vol. 82, no. 2, pp. 332-350.

#### Recommended Citation

Kurbanov, B. T.
(2020)
"BOUNDARY THEOREM OF MORERA IN THE SPACE OF RECTANGULAR MATRICES,"
*Karakalpak Scientific Journal*: Vol. 3
:
Iss.
1
, Article 3.

Available at:
https://uzjournals.edu.uz/karsu/vol3/iss1/3