Nonlinear singular integrals depending on two parameters (Q2752311)

Let \(G\) be a locally compact abelian group with the Haar measure \(dt\), and \(W\) be a set of indices with any topology. Let \(\{K_w\}_{w\in W}\) be a family of functions \(K_w: G\times \mathbb R\to \mathbb R\) with \(K_w(t,0)=0\). The authors discuss conditions guaranteeing the convergence of the integral operators, \(\int_G K_w(s-t, f(t)) dt \to f(s_0)\) as \((w,s)\to (w_0, s_0)\). Several examples are given. One of them is : \(K_n(t,u)=\frac{nu}{2}+\sin\frac{nu}{2}\) \((-\frac 1n \leq t\leq \frac 1n)\), \(=0\) \((t\in [-\pi, \pi]\setminus [-\frac 1n, \frac 1n])\), where \(G=[-\pi,\pi)\) and \(W=\mathbb N\).