Abstract:
In a Lebesgue space with weight $L^p_{\beta-{2}/{p}}(D)\ (1<p<\infty, 0<\beta<2),$ where $D$ is a finite singly connected domain of the complex plane bounded by a simple closed Lyapunov curve $\Gamma$ and containing the point $z = 0$, we consider a two-dimensional singular integral operator of the Mikhlin – Calderon – Zygmund type of the form \begin{equation} \notag \begin{split} & (Af)(z)\equiv a_0(z)f(z)+b_0(z)\overline{f(z)}+ \\ &+\iint_D\frac{\Omega_1(z,\theta)}{|\zeta-z|^2}f(\zeta)ds_\zeta+ \iint_D\frac{\overline{\Omega_2(z,\theta)}}{|\zeta-z|^2}\overline{f(\zeta)}ds_\zeta,\ \theta=\arg(\zeta-z). \end{split} \end{equation} Depending on the homotopy class $\mathfrak M_{\nu} (\nu=0,\pm 1,\ldots,\pm m)$ of the operator $A$, we establish effective necessary and sufficient conditions for the operator $A$ to be Noetherian in $L^p_{\beta-{2}/{p}}(D) (1<p<\infty, 0<\beta<2)$ and found formulas for calculating the index of an operator.
The results obtained are applied to the Dirichlet and Neumann problems for general elliptic systems of two equations with two higher-order independent variables.
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