On a coefficient in trace formulas for Wiener-Hopf operators

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: $$\mathrm{t}\mathrm{r}\left(g\right(W(a))-W(g\circ a))=\mathcal{B}(a;g),$$ where $\mathcal{B}(a;g)$ is given explicitly in terms of the functions $a$ and $g$. The paper analyses the coefficient $\mathcal B(a; g)$foraclassofnon-smoothfunctions$g$assumingthat$a$ is real-valued. A representative example of one such function is $g(t)=\mathrm{\mid }t{\mathrm{\mid }}^{\gamma }$ with some $\gamma \in (0,1]$.