Weak Orlicz spaces: some basic properties and their applications to harmonic analysis (Q365824)

Let \(\phi\) be a convex Orlicz function and let \((\Omega,\Sigma,\mu)\) be a complete, non-atomic and \(\sigma\)-finite measure space. The weak Orlicz space \(wL_\phi\) is defined as the space of all measurable functions \(f\) on \((\Omega,\Sigma,\mu)\) such that there exists a \(c>0\) for which \(\sup_{t>0} \phi(ct)\mu(|f|> t)<\infty\). The authors investigate the Hardy-Littlewood operator \[ Mf(x)= \sup_{x\in Q}\,{1\over|Q|} \int_Q |f(y)|\,dy \] in the spaces \(wL_\phi\). The main result consists in proving a theorem on Marcinkiewicz-type interpolation for operators in the spaces \(wL_\phi\). Applications of this result to problems of convolution, Paley-Wiener-type inequalities and Zygmund spaces are also given.