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Some general questions in the theory of the Riemann boundary problem

I. B. Simonenko


Abstract: In this paper we investigate the Riemann boundary problem
$$ \Phi^+(t)=G(t)\Phi^-(t)+g(t) $$
for $n$ pairs of functions. The solutions $\Phi^\pm$ are to belong to the classes $E_p^\pm$; the given function g belongs to the class $L_p$ $(1<p<\infty)$. We enlarge the class of coefficients $G$ for which the Noether theory remains valid. In the case $n=1$, $p=2$, necessary and sufficient conditions for Noetherianness are obtained. It is shown that the class of matrix-functions which admit factorization coincides with the class for which the Noether theory is valid. In the case $n=1$ it is shown that one of the defect numbers is zero.

UDC: 517.9

MSC: 30F20, 30E25, 28B20, 46E30, 47A56, 26B35

Received: 03.01.1968

 English version: Mathematics of the USSR-Izvestiya, 1968, 2:5, 1091–1099

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