Weighted estimates for a class of sublinear operators (Q403894)

Let \(M\) be the class of Lebesgue measurable functions defined on \((0,\infty)\), and let \(M^\ast :=\{f\in M:f\geq 0\}\). The paper under review is concerned with some sublinear operators with nonnegative measurable kernel functions defined on the cone \(M^\ast\). The authors prove some inequalities for such kind of operators when the kernel functions satisfy Oinarov's condition \(k(x,y)\approx k(x,z) + k(z,y)\) [\textit{R. Oĭ{n}arov}, Proc. Steklov Inst. Math. 204, 205--214 (1994); translation from Tr. Mat. Inst. Steklova 204, 240--250 (1993; Zbl 0883.47048)]. For a special case, their inequalities are reduced to the usual weighted inequalities with Volterra integral operators.