On interpolation of operator, which is the sum of weighted Hardy-Littlewood and Cesaro mean operators

Abstract

It is proved that operators, which are the sum of weighted Hardy-Littlewood $$$\int\limits_0^1 f(xt) \psi(t) dt$$$ and Cesaro $$$\int\limits_0^1 f(\frac{x}{t}) t^{-n} \psi(t) dt$$$ mean operators, are limited on Lorentz spaces $$$\Lambda_{\varphi, a} (\mathbb{R})$$$, if the functions $$$f(x) \in \Lambda_{\varphi, a}(\mathbb{R})$$$ satisfy the condition $$$|f(-x)| = |f(x)|$$$, $$$x > 0$$$, for such non-increasing semi-multiplicative functions $$$\psi$$$, for which the next conditions are satisfied: $$$\frac{M_1}{\psi(t)} \leqslant \psi(\frac{1}{t}) \leqslant \frac{M_2}{\psi(t)}$$$, for all $$$0 < t \leqslant 1$$$; at some $$$0 < \varepsilon < \frac{1}{2}$$$, $$$0 < \delta < \frac{1}{2}$$$ functions $$$\psi(t) t^{1-\varepsilon}$$$, $$$\psi(\frac{1}{t}) t^{-\delta}$$$ do not decrease monotonically and functions $$$\psi(t) t$$$, $$$\psi (\frac{1}{t})$$$ are absolutely continuous. Also, there are proved sufficient conditions that the operators, which are the sum of weighted Hardy-Littlewood and Cesaro mean operators, when $$$\psi(t) = t^{-\alpha}$$$, where $$$\alpha \in (0, \frac{1}{2})$$$, on Lorentz spaces $$$\Lambda_{\varphi, a}(\mathbb{R})$$$, if the functions $$$f(x) \in \Lambda_{\varphi, a}(\mathbb{R})$$$ satisfy the condition $$$|f(-x)| = |f(x)|$$$, $$$x > 0$$$.