Weak type and restricted weak type \((p,p)\) operators in Orlicz spaces.

Abstract

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.