Weighted Lorentz spaces with iterated norms and optimal domains of Hardy operators

Abstract

We show the weighted norm inequalities for integral operators and the weight characterization for the Hardy inequality .Banach space properties of Lebesgue space of real functions of a vector measure ,and optimal Sobolev imbeddings involving rearrangement –invariant quasinorms ,are considered .We also show the optimal domains for the kernel operator of Sobolev inequality and Lebesgue spaces of vector measures on rings .We discuss the iterated norms and norm inequalities in multidimensional Lorentz spaces. Hardy operator and transforms of weights are also considered .We study weak type weights and new Lorentz spaces for weak-type Hardy inequalities .We investigate the general Banach lattices with the Fatou property and the optimal domains for the Hardy operator .New characterized results are studied.