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

B.I. Peleshenko; T.N. Semirenko

doi:10.15421/241809

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

B.I. Peleshenko (Dnipro State Agrarian and Economic University)
T.N. Semirenko (Dnipro State Agrarian and Economic University)

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.

Keywords

the Lorentz space; operators interpolation; slowly varying functions; nonincreasing permutation of functions; index of tension

MSC 2020

41A05; 47A58; 46E30; 46A32

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References

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