Wiener-Hopf integral operators with $PC$ symbols on spaces with Muckenhoupt weight

Abstract

We describe the spectrum and the essential spectrum and give an index formula for Wiener-Hopf integral operators with piecewise continuous symbols on the space $L^P({\mathbb{R}}_{+},\omega )$ with a Muckenhoupt weight $\omega$. Our main result says that the essential spectrum is a set resulting from the essential range of the symbol by joining the two endpoints of each jump by a certain sickle-shaped domain, whose shape is completely determined by the value of $p$ and the behavior of the weight $\omega$ at the origin and at infinity.