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

Calderon A.P. "Spaces between $$$L^1$$$ and $$$L^{\infty}$$$ and theorem of Marcinkiewich", Studia Math., 1966; 26: pp. 273-299.

Boyd D.W. "Indices of function spaces and their relationship to interpolation", Canad. Math. J., 1969; 21: pp. 1245-1254.

Sharpley R.C. "Spaces $$$\Lambda_{\alpha}(X)$$$ and interpolation", J. Functional Anal., 1972; 11(4): pp. 479-513.

Dmitriev V., Krein S.G. "Interpolation of operator weak type", Analysis Math., 1978; 4(2): pp. 83-99.

Bennett C., Rudnick K. On Lorentz-Zygmund spaces, Panstw. wydawn. nauk., 1980; 73 p.

Bennett C., Charpley R. "Interpolation of Operators", Pure Appl. Math., 1988; 129: pp. 1-469.

Krein S.G., Petunin Yu.I., Semyonov Ye.M. Interpolation of linear operators, Nauka, 1978.

Pavlov Ye.A. "On Calderon operator", Analysis Math., 1978; 4(2): pp. 117-124.

Peleshenko B.I. "On operators of weak type", Works of Ukrainian Math. Congress-2001. Functional Analysis, Kyiv, 2002; pp. 234-244.

Burenkov V.I., Goldman M.L. "Calculations of the norm of positive operator on the cone of monotonic functions", Trudy Matem. in-ta rAN, 1995; 210: pp. 65-89.

Peleshenko B.I. "Interpolation of operators of weak type $$$(\phi_0, \psi_0, \phi_1, \psi_1)$$$ in Lorentz spaces", Ukrainian Math. J., 2005; 57(11): pp. 1490-1507.

Peleshenko B.I. "Interpolation of operators of weak types in borderline cases", Res. Math., 2006; 13: pp. 71-83.