Singular integral operators. The case of an unlimited contour

Let \(\Gamma\)be a closed or unclosed unlimited contour, a shift \(\alpha(t)\) maps homeomorphicly the contour \(\Gamma\) onto itself with preserving or reversing the direction on \(\Gamma\) and also satisfies the conditions: for some natural \(n\geq2\), \(\alpha_n(t)\equiv t\), and \(\alpha_j(t)\n...

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Bibliographic Details
Main Author: V. Neaga
Format: Article
Language:English
Published: Publishing House of the Romanian Academy 2005-08-01
Series:Journal of Numerical Analysis and Approximation Theory
Subjects:
Online Access:https://www.ictp.acad.ro/jnaat/journal/article/view/802
Description
Summary:Let \(\Gamma\)be a closed or unclosed unlimited contour, a shift \(\alpha(t)\) maps homeomorphicly the contour \(\Gamma\) onto itself with preserving or reversing the direction on \(\Gamma\) and also satisfies the conditions: for some natural \(n\geq2\), \(\alpha_n(t)\equiv t\), and \(\alpha_j(t)\not\equiv t\) for \(1\leq j<n\). In this work we study subalgebra \(\Sigma\) of algebra\(L(L_p(\Gamma,\rho))\), which contains all operators of the form\[\left (M \varphi \right) (t) = \sum_{k=0}^{n-1} \bigg \{a_k (t) \varphi (\alpha_k (t)) + \tfrac{b_k(t)}{\pi {\rm i} } \int_{\Gamma} \tfrac{\varphi ( \tau )}{\tau - \alpha_k (t)} d \tau \bigg \}\]with piecewise-continuous coefficients. The existence of such an isomorphism between \(\Sigma\) and some algebra \(\frak A\) of singular operators with Cauchy kernel that an arbitrary operator from \(\Sigma\) and its image are Noetherian or not Noetherian simultaneously is proved. It allows to introduce the concept of a symbol for all operators from \( \Sigma \) and, using the known results for algebra \( \frak A \), in terms of a symbol to receive conditions of Noetherian property.
ISSN:2457-6794
2501-059X