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

摘要

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 ({pin(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 (({ain PSO^diamond})) and all convolution operators W 0(b) (({bin PSO_{p,w}^diamond})), where ({PSO^diamondsubset L^infty(mathbb{R})}) and ({PSO_{p,w}^diamondsubset 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{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 ({Ainmathfrak{U}_{p,w}}) in terms of their Fredholm symbols. In four partial cases we obtain for ({mathfrak{U}_{p,w}}) more effective results.