Banach Algebras of Convolution Type Operators with PQC Data

Banach algebras \({\mathfrak A}_{\mathbb {R},p,w}\) of convolution type operators with piecewise quasicontinuous data that have piecewise slowly oscillating asymptotics at infinity are studied on weighted Lebesgue spaces \(L^p(\mathbb {R},w)\) with \(p\in (1,\infty )\) and a subclass of Muckenhoupt weights w. A Fredholm symbol calculus for the algebra \({\mathfrak A}_{\mathbb {R},p,w}\) is constructed and a Fredholm criterion for the operators \(A\in {\mathfrak A}_{\mathbb {R},p,w}\) is established.