Singular integral operators with fixed singularities on weighted Lebesgue spaces (Q1880349)

The authors construct a symbol calculus for singular integral operators with fixed singularities at the end points of smooth contours, on weighted Lebesgue spaces with slowly oscillating Muckenhoupt weights. One of the main results of the paper is to prove that operators with fixed singularities on weighted Lebesgue spaces with slowly oscillating Muckenhoupt weights belong to a Banach algebra generated by the Cauchy singular operator and the operators of multiplication by continuous functions. This result is proved from the theory of strongly continuous semigroups of closed linear operators using Balakrishnan's formula. The well-known symbol calculus for the algebra of singular integral operators with continuous coefficients on weighted Lebesgue spaces allows one to get a Fredholm criterion and an index formula for the above mentioned operators with fixed singularities. The main point is the construction of a Fredholm theory for the operator \(A=\rho K_{\eta,h} \rho^{-1}I\), defined on the space \(L^p(J,w)\), \(1<p<\infty\), where \(K_{\eta,h}\) is given by \[ (K_{\eta,h}f)(t)=\frac{1}{\pi i} \int^{}_{J}\frac{\eta'(x)}{\eta(x)-h(t)}f(x)dx,\quad t\in J=[0,1], \] the functions \(\eta,h\) being orientation preserving diffeomorphisms of the segment \(J\) onto smooth arcs (either \(\eta=h\) identically on \(J\) or \(\eta(x)=h(t)\) only for \(x=t=0\) and/or \(x=t=1\)), and \(\rho(t)=t^\gamma(1-t)^\delta\) with \(-\upsilon_0^-(\omega)<\gamma<1-\upsilon_0^+(\omega)\), \(-\upsilon_1^-(\omega)<\delta<1-\upsilon_1^+(\omega)\), where \(\upsilon_t^{\pm}(\omega)\), \(t=1,2\), are quantities associated to the slowly oscillating Muckenhoupt weights \(\omega\in A_p(J)\).