Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. II: Local Spectra and Fredholmness

Abstract

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 \({\mathfrak{U}_{p,w}}\) of \({\mathcal{B}_{p,w}}\) generated by all multiplication operators aI (\({a\in PSO^\diamond}\)) and all convolution operators W0(b) (\({b\in PSO_{p,w}^\diamond}\)), where \({PSO^\diamond\subset L^\infty(\mathbb{R})}\) and \({PSO_{p,w}^\diamond\subset M_{p,w}}\) are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of \({\mathbb{R}\cup\{\infty\}}\) , and Mp,w is the Banach algebra of Fourier multipliers on \({L^p(\mathbb{R},w)}\) . Under some conditions on the Muckenhoupt weight w, using results of the local study of \({\mathfrak{U}_{p,w}}\) obtained in the first part of the paper and applying the theory of Mellin pseudodifferential operators and the two idempotents theorem, we now construct a Fredholm symbol calculus for the Banach algebra \({\mathfrak{U}_{p,w}}\) and establish a Fredholm criterion for the operators \({A\in\mathfrak{U}_{p,w}}\) in terms of their Fredholm symbols. In four partial cases we obtain for \({\mathfrak{U}_{p,w}}\) more effective results.