On integral operators of Wiener-Hopf type with piecewise difference kernels (Q1336222)

The author considers the convolution-type operator \[ R\varphi:= \varphi(t)+ \sum_{s=1}^ n \chi_ s(t) \int_ 0^ t a_ s (t-\tau) \varphi(\tau) d\tau, \qquad t>0, \] in the space \(L_ p (0,\infty)\), \(\chi_ s(t)\) being characteristic functions of the intervals \((\alpha_{s-1}, \alpha_ s)\), \(s=1,\dots, n\); \(0< \alpha_ 0< \alpha_ 1< \dots< \alpha_{n-1}< \alpha_ n= \infty\) and it is assumed that \(a_ s(t)\in L_ 1 (0,\infty)\). Let \[ A_ s(x) =1+ \int_ 0 a_ s(t) \exp (ixt) dt, \qquad s=1,\dots, n, \] and let \(A_ n(x) \neq 0\), \(x\in R^ 1\). Let also \(\mu_ j\), \(j=1,\dots, k\), be the distinct common zeros of all the \(A_ 1 (z), \dots, A_ n(z)\) in the upper half plane and \(\rho_ j\) be the common multiplicity of these zeroes. The main result is a theorem stating that 1) \(\dim\ker R= N:= \rho_ 1+ \dots+ \rho_ k\) and \(\ker R\) is defined by the basis \((t^ 1 \exp (-\mu_ j (t))\), \(j=1,\dots, k\); \(l= 1,\dots, \rho_ j-1\); 2) \(\text{dim coker } R= N_ n- N\), \(N_ n=- \text{ind } R\). Reviewer's remark. Both statements are erroneous, since \(\text{Im } \mu_ j>0\), so that \(t^ 1\exp (-i \mu_ j t)\not\in L_ p (0,\infty)\) and \(\dim\ker R- \text{dim coker } R\) should be equal to \(\text{ind } R\).