Algebras of convolution type operators with piecewise slowly oscillating data. II: Local spectra and Fredholmness
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. The authors study the Banach subalgebra \({\mathfrak{A}_{p,w}}\) of \({\mathcal{B}_{p,w}}\) generated by all multiplication operators \(aI\) \(({a\in PSO^\diamond})\) and all convolution operators \(W ^{0}(b)\) \(({b\in PSO_{p,w}^\diamond})\), 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 \(M _{p,w }\) 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 \({\mathfrak{A}_{p,w}}\) obtained in the Part I [ibid. 74, No.~3, 377--415 (2012; Zbl 1275.47103)] and applying the theory of Mellin pseudodifferential operators and the two idempotents theorem, the authors construct a Fredholm symbol calculus for the Banach algebra \({\mathfrak{A}_{p,w}}\) and establish a Fredholm criterion for the operators \({A\in\mathfrak{A}_{p,w}}\) in terms of their Fredholm symbols. In four partial cases, they obtain more effective results for \({\mathfrak{A}_{p,w}}\).
scientific article; zbMATH DE number 6153959
scientific article