Algebras of Singular Integral Operators with Piecewise Continuous Coefficients on Reflexive Orlicz Spaces

Abstract

AbstractWe consider singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces Lm(σ) which are generalizations of the Lebesgue spaces LP(σ), 1 < p < ∞. We suppose that σ belongs to a large class of Carleson curves, including curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. For the singular integral operator associated with the Riemann boundary value problem with a piecewise continuous coefficient G, we establish a Fredholm criterion and an index formula in terms of the essential range of G complemented by spiralic horns depending on the Boyd indices of LM(σ) and contour properties. Our main result is a symbol calculus for the closed algebra of singular integral operators with piecewise continuous matrix ‐ valued coefficients on LMn(σ).