Extension of the definition domain of singular integral operators with degenerating symbol (Q1022236)

The paper is devoted to the normalization problem for singular integral operators of the form \(aP_++bP_-+K\) and \(aP_++bUP_++cP_-+dUP_-+K\) acting on vector Lebesgue spaces \(L_p^m(\Gamma)\). Here, \(\Gamma\) is a simple closed Lyapunov curve for the first operator and the unit circle for the second operator; \(P_\pm\) are the projections associated with the Cauchy singular integral operator; \(U\) is the weighted shift operator given by \((U\varphi)(t)=w(t)\varphi(\alpha(t))\), where \(\alpha(t)=(t-\beta)/(\overline{\beta}t-1)\), \(w(t)=i\sqrt{|\beta|^2-1}/(\overline{\beta}t-1)\), and \(|\beta|>1\); \(K\) is a compact operator; and \(a,b,c,d\in L_\infty^{m\times m}(\Gamma)\). These operators are not normally solvable if coefficients are allowed to degenerate. Under some regularity conditions on the coefficients, the authors construct a different domain for these operators (keeping the target space \(L_p^m(\Gamma)\) unchanged) so that, considered on this domain, the redefined operators become normally solvable.