Pseudodifferential operators approach to singular integral operators in weighted variable exponent Lebesgue spaces on Carleson curves
The paper presents general results concerning one-dimensional singular integral operators, i.e., operators with Cauchy kernel, on Carleson curves \(\Gamma\). The functional frame is given by Lebesgue spaces \(L^{p(\cdot)}(\Gamma, w)\) with variable exponent \(p= p(x)\), \(x\in\Gamma\), and weights \(w\) having a finite set of oscillation singularities. The authors adopt the modern techniques for pseudodifferential and Mellin type operators, and obtain precise criteria for Fredholmness. An important tool in the proofs is represented by the local principle of \textit{I. B. Simonenko} [Izv. Akad. Nauk SSSR, Ser. Mat. 29, 567--586, 757--782 (1965; Zbl 0146.13101)].
scientific article; zbMATH DE number 5898353
scientific article