Algebras of Convolution Type Operators with Piecewise Quasicontinuous and Piecewise Slowly Oscillating Data on Weighted Lebesgue Spaces

Let \(\mathcal {B}_{p,w}\) be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space \(L^p(\mathbb {R},w)\) , where \(p\in (1,\infty )\) and w is a Muckenhoupt weight. We study the Banach subalgebra \(\mathfrak {A}_{p,w}\) of \(\mathcal {B}_{p,w}\) generated by all multiplication operators aI ( \(a\in PQC\) ) and all convolution operators \(W^0(b)\) ( \(b\in PSO_{p,w}^\diamond \) ), where \(PQC\subset L^\infty (\mathbb {R})\) is the \(C^*\) -algebra of piecewise quasicontinuous functions, \(PSO_{p,w}^\diamond \subset M_{p,w}\) is the Banach algebra of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of \(\mathbb {R}\cup \{\infty \}\) , and \(M_{p,w}\) is the Banach algebra of Fourier multipliers on \(L^p(\mathbb {R},w)\) . For any \(p\in (1,\infty )\) and any Muckenhoupt weight w, we study the Fredholmness in the Banach algebra \({\mathcal Z}_{p,w}\subset \mathfrak {A}_{p,w}\) generated by the operators \(aW^0(b)\) with quasicontinuous functions \(a\in QC\) and slowly oscillating functions \(b\in SO^\diamond _{p,w}\) . Applying the Allan-Douglas local principle and the results for the algebra \({\mathcal Z}_{p,w}\) , we study the Fredholmness of operators \(A\in \mathfrak {A}_{p,w}\) .