RO  EN
IMCS/Publications/BASM/Issues/BASM n.3 (31), 1999/

Singular integral operators. I. Case of an unlimited contour. (English)

Authors: Neaga V.

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 $$\left (M \varphi \right) (t) = \sum_{k=0}^{n-1}\left \{a_k (t) \varphi (\alpha_k (t)) + \frac{b_k(t)}{\pi i } \int\limits_{\Gamma} \frac{\varphi ( \tau )}{\tau -\alpha_k (t)} d \tau \right \}$$ 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.}

Universitatea de Stat din Moldova
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