Toeplitz and singular integral operators on Carleson curves with logarithmic whirl points

Abstract

We consider Toeplitz operators with piecewise continuous symbols and singular integral operators with piecewise continuous coefficients onL p (Γ,w) where 1<p<∞,w is a Muckenhoupt weight and Γ belongs to a large class of Carleson curves. This class includes curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. Our main result says that the essential spectrum of a Toeplitz operator is obtained from the essential range of its symbol by joining the endpoints of each jump by a certain spiralic horn, which may degenerate to a usual horn, a logarithmic spiral, a circular arc or a line segment if the curve Γ and the weightw behave sufficiently well at the point where the symbol has a jump. This result implies a symbol calculus for the closed algebra of singular integral operators with piecewise continuous coefficients onL p (Γ,w).