On boundedness of operators of weak type $$$(\varphi_0, \psi_0, \varphi_1, \psi_1)$$$ in Lorentz spaces in limit cases

B.I. Peleshenko

doi:10.15421/240716

On boundedness of operators of weak type $$$(\varphi_0, \psi_0, \varphi_1, \psi_1)$$$ in Lorentz spaces in limit cases

B.I. Peleshenko (Dnipropetrovsk State Agrarian University)

Abstract

We prove theorems on boundedness of operators of weak type $$$(\varphi_0, \psi_0, \varphi_1, \psi_1)$$$ from Lorentz space $$$\Lambda_{\varphi,a}(\mathbb{R}^n)$$$ to $$$\Lambda_{\varphi,b}(\mathbb{R}^n)$$$ in “limit” cases, when one of functions $$$\varphi(t) / \varphi_0(t)$$$, $$$\varphi(t) / \varphi_1(t)$$$ slowly changes at zero and at infinity.

Full Text:

PDF (Русский)

References

Krein S.G., Petunin Yu.I., Semyonov Ye.M. Interpolation of linear operators, Nauka, Moscow, 1978; 400 p. (in Russian)

Peleshenko B.I. "On operators of weak type $$$(\phi_0, \psi_0, \phi_1, \psi_1)$$$", Works of Ukrainian Math. Congress-2001. Functional Analysis, Kyiv, 2002; pp. 234-244. (in Russian)

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. (in Russian) doi:10.1007/s11253-006-0027-3

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

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

Boyd D.W. "Indices of function spaces and their relationship to interpolation", Canad. Math. J., 1969; 21: pp. 1245-1254. doi:10.4153/CJM-1969-137-x

Calderon A.P. "Spaces between $$$L^1$$$ and $$$L^{\infty}$$$ and theorem of Marcinkiewicz", Studia Math., 1965; 26(3): pp. 273-299. doi:10.4064/sm-26-3-301-304

Sharpley R.C. "Spaces $$$\Lambda_{\alpha}(X)$$$ and interpolation", J. Functional Anal., 1972; 11(4): pp. 479-513. doi:10.1016/0022-1236(72)90068-7