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2002/2003 Weak type and restricted weak type \((p,p)\) operators in Orlicz spaces.
B. Bongioanni, L. Forzani, E. Harboure
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Real Anal. Exchange 28(2): 381-394 (2002/2003).


Let \((\Omega,\mu)\) be a finite measure space, \(\Phi(t)=\int_{0}^{t} a(s)\, ds\) and \(\Psi(t)=\int_{0}^{t} b(s)\, ds\), where \(a\) and \(b\) are positive continuous functions defined on \([0,\infty)\). Consider the associated Orlicz spaces \(L^{\Phi}(\Omega)\) and \(L^{\Psi}(\Omega)\). In this paper we find a relationship between \(a\) and \(b\) to assure that \(T\), a sublinear and positive homogeneous operator of restricted weak type \((p,p)\) and of type \((\infty,\infty)\), maps \(L^{\Psi}(\Omega)\) into \(L^{\Phi}(\Omega)\). If the two Orlicz spaces are normable, our results imply the continuity of \(T\). This relation between \(a\) and \(b\) is sharp since it is shown to be necessary for operators like the one side maximal operators related to the Ces\`aro averages.


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B. Bongioanni. L. Forzani. E. Harboure. "Weak type and restricted weak type \((p,p)\) operators in Orlicz spaces.." Real Anal. Exchange 28 (2) 381 - 394, 2002/2003.


Published: 2002/2003
First available in Project Euclid: 20 July 2007

zbMATH: 1082.46021
MathSciNet: MR2009761

Primary: 42B25 , 46E30

Keywords: interpolation , maximal function , Orlicz space

Rights: Copyright © 2002 Michigan State University Press

Vol.28 • No. 2 • 2002/2003
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