On a coefficient in trace formulas for Wiener–Hopf operators

Abstract

Let $a=a(\xi ),\xi \in \mathbb{R},$ be a smooth function quickly decreasing at infinity. For the Wiener–Hopf operator $W(a)$ with the symbol $a$, and a smooth function $g:\mathbb{C}\to \mathbb{C}$, H. Widom in 1982 established the following trace formula:

where $\mathcal{B}(a;g)$ is given explicitly in terms of the functions $a$ and $g$. The paper analyses the coefficient $\mathcal{B}(a;g)$ for a class of non-smooth functions $g$ assuming that $a$ is real-valued. A representative example of one such function is $g(t)=|t{|}^{\gamma }$ with some $\gamma \in (0,1]$.