Singular integral equations on the real line with a fractional-linear Carleman shift (Q2774467)

Singular integral operators of the following form are considered NEWLINE\[NEWLINE(K\varphi)(x)= a(x)\varphi(x)+ b(x)\varphi(\tau(x))+ c(x)(S\varphi)(x)+ d(x)(S\varphi)(\tau(x)),NEWLINE\]NEWLINE where \(x\) is real, the coefficients are continuous, \(S\) is the Hilbert transform and the Carleman shift \(\tau\) is an involutive fractional linear transformation on \(\mathbb{R}\). Fredholm properties of the operator \(K\) are considered in weight spaces of the form NEWLINE\[NEWLINEL^\gamma_p(\mathbb{R})= \Biggl\{\varphi: \int^{+\infty}_{-\infty}|x-\delta|^\gamma |\varphi(x)|^p dx< \infty\Biggr\},NEWLINE\]NEWLINE where \(\delta\) is the singular point of the shift, \(-1< \gamma< p-1\) and \(1< p< \infty\). In particular, the Fredholm index is calculated. The case \(\gamma= {p\over 2}-1\) had previously been studied by the authors and exhibits relatively straightforward Fredholm properties. However, in the general case (which may include the unweighted case \(\gamma= 0\)), some additional conditions are required. These arise because the operator \(K\) reduces not just to a singular operator as in the case \(\gamma= {p\over 2}-1\), but to such an operator perturbed by an integral operator with homogeneous kernel.