Riemann boundary problem with infinite index of logarithmic order on a spiral-form contour. II

We study the inhomogeneous Riemann boundary problem with infinite index of logarithmic order on an open rectifiable spiral-form Jordan contour where the influence of the contour on the solvability of the problem is comparable with the influence of the argument of its coefficient. A solution of the problem is constructed explicitly in the class of functions admitting weak power singularities at the ends of a line of conjugacy.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1672–1681, December, 1990.     

Riemann boundary problem with index plus-infinity on a rectifiable curve

A piecewise-continuous Riemann boundary problem with index plus-infinity on a closed rectifiable Jordan curve is studied; here the index of the problem takes into account the integral influence on its solvability of the argument and modulus of the coefficient of the problem and also the properties of the line of conjugation. One permits discontinuities of the second kind in the logarithm of the coefficient and the free term of the problem. The solution of the problem is constructed explicitly in the class of functions admitting a weak-power singularity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1204–1213, September, 1990.     

A new approach to solving homogeneous Riemann boundary-value problem on a ray with infinite index

We consider a Riemann boundary-value problem with infinite Gakhov’s index. The boundary data are defined on positive ray of the real axis. We solve the problem by means of removal of singularity of boundary data at the infinity. This approach is analogous to Gakhov’s method of elimination of singularities in the problemswith finite indices, but we use another eliminating factors.     

The Riemann problem on a finite-sheeted Riemann surface of infinite genus

LetR be the Riemann surface of the functionu(z) specified by the equationu ⁿ=P(z) withn ε ℕ,n ≥ 2, andz ε ℂ, whereP(z) is an entire function with infinitely many simple zeros. OnR, the Riemann boundary-value problem for an arbitrary piecewise smooth contour Γ is considered. Necessary and sufficient conditions for its solvability are obtained, and its explicit solution is constructed. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 25–35, January, 2000.     

Analytic self-mappings of infinite order of Riemann surfaces

Sans résumé Supported in part by the National Science Foundation.     

The Cauchy problem for evolution equations with the Bessel operator of infinite order. I

We establish necessary and sufficient conditions under which the Bessel operator of infinite order is bounded in certain spaces. We study properties of the Bessel transformations of distributions from these spaces, those of convolutions, convolutors, and multiplicators.     

Riemann boundary-value problem with index minus infinity on a rectifiable curve

The author investigates the piecewise-continuous Riemann boundary-value problem with index minus infinity on a closed rectifiable Jordan curve; the index of the problem is a measure of the integral effect exerted on the solvability of the problem by the argument and modulus of the coefficient, and also by the properties of the junction curve. Discontinuities of the second kind are admitted in the logarithm of the coefficient and in the free term. The solution of the problem is constructed explicitly in the class of functions having a weak power singularity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1350–1356, October, 1990.