Optimal Exponents for Hardy–Littlewood Inequalities and Riesz–Morrey–Hardy–Sobolev Spaces and Potentials

Abstract

The study started by harmonic maps, Morrey potentials and Restrictions of Riesz – Morrey – Hardy inequalities, Morrey spaces in harmonic analysis with generalized local Morrey spaces and the fractional integral operators with rough kernel. Remarks and an interpolation approach to Hardy - Littlewood inequalities for norms of operators on sequence spaces are presented. Optimal and optimal exponents for Hardy – Littlewood type in equalities by blocks, for polynomials, multilinear operators, for m-linear forms on l_pspaces and for m-linear operators are given. The decomposition and generalized Hardy – Morrey spaces with trace law for Hardy – Morrey – Sobolev spaces are established. We find the singular integral operator and Hardy – Morrey space estimates for multilinear operators and Bilinear estimate in weak – Morrey spaces and uniqueness for Navier – Stokes equations.