Singular integral operators. The case of an unlimited contour

V. Neaga

doi:10.33993/jnaat342-802

Authors

V. Neaga State University of Moldova, Chisinau, Republic of Moldova

DOI:

https://doi.org/10.33993/jnaat342-802

Keywords:

Lyapunov closed curves, Noetherian singular integral, piecewise Lyapunov contour, singular integral equations, singular operators with Cauchy kernel singular operators with shift

Abstract views: 206

Abstract

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.

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