On weighted estimates for a class of integral operators
Let \(\varphi(t)\), \(t\in (0,1)\), be a measurable function satisfying the following conditions: a) \(\varphi(t) \geq 0\) and \(\varphi(t)\) is nonincreasing; b) \(\varphi (t_ 1+ t_ 2)\leq D(\varphi(t_ 1)+ \varphi(t_ 2))\), \(0<t_ 1,t_ 2< 1\), where \(D\) is independent of \(t_ 1\), \(t_ 2\). For the integral operator \[ (T_ \varphi f) (x)= \int_ 0^ x \varphi(t/x) f(t) dt \] weighted norm inequalities of the form \[ \| u T_ \varphi f \|_{L^ q (0,\infty)} \leq c\| vf \|_{L^ p (0,\infty)} \] are studied in the case \(0<q<p <\infty\), \(p>1\). Moreover, necessary and sufficient conditions for the operator \(T_ \varphi\) to be compact are given in the case \(1<p, q<\infty\).
scientific article; zbMATH DE number 652187
scientific article