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 $|f(-x)| = |f(x)|$, $x > 0$.