On interpolation of operators of weak type $$$(\phi_0, \psi_0, \phi_1, \psi_1)$$$ in Lorentz spaces in borderline cases

Abstract

The quaslinear operators of weak type $$$(\phi_0, \psi_0, \phi_1, \psi_1)$$$, analogs of the Calderon, Bennett operators in the case of concave and convex functions $$$\phi_0(t)$$$, $$$\psi_0(t)$$$, $$$\phi_1(t)$$$, $$$\psi_1(t)$$$ are considered. The theorems of interpolation of these operators from the Lorentz space $$$\Lambda_{\psi, b}(\mathbb{R}^n)$$$ into the space $$$\Lambda_{\psi, a}(\mathbb{R}^n)$$$ in cases when $$$0 < b \leqslant a \leqslant 1$$$ and relation of function $$$\phi^{\frac{1}{b}}(t)$$$ to one of functions $$$\phi_1(t)$$$, $$$\phi_2(t)$$$ is slowly varying function are proved.