The boundedness of the Cauchy singular integral operator in weighted Besov type spaces with uniform norms

Abstract

The mapping properties of the Canchy singular integral operator with constant\ud coefficients are studied in couples of spaces equipped with weighted uniform\ud norms. Recently weighted Besov type spaces got more and more interest in\ud approximation theory and, in particular, in the numerical analysis of polynomial\ud approximation methods for Cauchy singular integral equations on an interval. In\ud a scale of pairs of weighted Besov spaces the authors state the boundedness and\ud the invertibility of the Canchy singular integral operator. Such result was not\ud expected for a long time and it will affect further investigations essentially.\ud The technique of the paper is based on properties of the de la Vall'èe Poussin\ud operator constructed with respect to some Jacobi polynomials

Mapping Properties of Some Singular Operators in Besov Type Subspaces of C( − 1, 1)

“…(3.4) The operators A and have been extensively studied in [21] and, for the sake of completeness, we mention the following results.…”

“…In the literature there exist partial results on this topic [13] [5] [6] [19] [20] [21]. Only recently the case A α,β with α + β = 0, 0 < |α| < 1 has been studied in detail in [21].…”

“…Only recently the case A α,β with α + β = 0, 0 < |α| < 1 has been studied in detail in [21]. The other two cases α + β ∈ {−1, 1}, 0 < |α|, |β| < 1 are still open problems.…”

Numerical Methods for Cauchy Singular Integral Equations in Spaces of Weighted Continuous Functions

“…To prove the proposition we need some preliminary results. First of all since [24] (Dφ)v 0,α ∞ ≤ C φv 13) …”

“…In this paper we consider D as a mapping from Z r (v α,0 ) into Z r (v 0,α ), r > 0. In [24] the authors extensively studied this operator. For the convenience of the reader we recall here the following results…”