Wiener-Hope equations with the transmission property

Conditions on the kernel of the classical Wiener-Hopf equation are obtained which provide the same smoothness of any solution in terms of its inclusion in different spaces of smooth functions such as as the function in the right side of the equation.     

Microbiological transformations of 9-amino-1,2,3,4,5,6,7,8-octahydroacridine

Chemistry Faculty, M. V. Lomonosov Moscow State University, Moscow 199899. Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 5, pp. 712–713, May, 1994. Original article submitted April 25, 1994     

On the Prandtl Equation

The unique solvability of the airfoil (Prandtl) integro-differential equation on the semi-axis ⁺ = [0, ) is proved in the Sobolev space W _p ¹ and Bessel potential spaces H _p ^s under certain restrictions on p and s.     

Finite Interval Convolution Operators with Transmission Property

We study convolution operators in Bessel potential spaces and (fractional) Sobolev spaces over a finite interval. The main purpose of the investigation is to find conditions on the convolution kernel or on a Fourier symbol of these operators under which the solutions inherit higher regularity from the data. We provide conditions which ensure the transmission property for the finite interval convolution operators between Bessel potential spaces and Sobolev spaces. These conditions lead to smoothness preserving properties of operators defined in the above-mentioned spaces where the kernel, cokernel and, therefore, indices do not depend on the order of differentiability. In the case of invertibility of the finite interval convolution operator, a representation of its inverse is presented in terms of the canonical factorization of a related Fourier symbol matrix function.     

On Estimates of the Boltzmann Collision Operator with Cut-off

We present new estimates of the Boltzmann collision operator in weighted Lebesgue and Bessel potential spaces. The main focus is put on hard potentials under the assumption that the angular part of the collision kernel fulfills some weighted integrability condition. In addition, the proofs for some previously known -estimates have been considerably shortened and carried out by elementary methods. For a class of metric spaces, the collision integral is seen to be a continuous operator into the same space. Furthermore, we give a new pointwise lower bound as well as asymptotic estimates for the loss term without requiring that the entropy is finite.     

Singular Integral Operators on an Open Arc in Spaces with Weight

The aim of this work is to study a singular integral operator ${\mathbf{A}=aI+bS_\Gamma}$ A = a I + b S Γ with the Cauchy operator S _Γ (SIO) and Hölder continuous coefficients a, b in the space ${\mathbb{H}^0_\mu(\Gamma,\rho)}$ H μ 0 ( Γ , ρ ) of Hölder continuous functions with an power “Khvedelidze” weight. The underlying curve is an open arc. It is well known, that such operator is Fredholm if and only if, along with the ellipticity condition ${a^2(t)-b^2(t)\not=0,\, t\in\Gamma}$ a 2 ( t ) - b 2 ( t ) ≠ 0 , t ∈ Γ , the “Gohberg–Krupnik arc condition” is fulfilled (see Duduchava, in Dokladi Akademii Nauk SSSR 191:16–19, 1970). Based on the Poincare–Beltrami formula for a composition of singular integral operators and the celebrated Muskhelishvili formula describing singularities of Cauchy integral, the formula for a composition of weighted singular integral operators is proved. Using the obtained composition formula and the localization, the Fredholm criterion of the SIO is derived in a natural way, by looking for the regularizer of the operator ${\mathbf{A}}$ A and equating to 0 non-compact operators. The approach is space-independent and this is demonstrated on similar results obtained for SIOs with continuous coefficients in the Lebesgue spaces with a “Khvedelidze” weight ${\mathbb{L}_p(\Gamma,\rho)}$ L p ( Γ , ρ ) , investigated earlier by Gohberg and Krupnik (Studia Mathematica 31:347–362, 1968; One Dimensional Singular Integral Operators II, Operator Theory, Advances and Applications, vol. 54, chapter IX, 1979) with a different approach.