Bounded operators on weak Hardy spaces and applications
The author considers Vilenkin systems and martingales with respect to the sequence of \(\sigma\)-algebras generated by a Vilenkin system. A new concept of atoms, the so-called weak atom is introduced, and the martingales from the weak Hardy space \(wH_p\), \(0<p<\infty\), are decomposed into the sum of weak atoms. This atomic decomposition is used for some applications. Some sufficent conditions for a sublinear operator \(T\) to be bounded from \(wH_p\) to \(wL_p\) are given as well as a weak version of the Hardy-Littlewood inequality is obtained. Vilenkin-Fourier series are considered and by certain assumptions almost everywhere convergence of the Cesàro means of a function \(f\in L_1\) is proved.
scientific article; zbMATH DE number 1261214