Sharp Hardy-Lorentz Spaces for B-Valued Martingales and Inequalities on Solid Torus with Limited Ranges of Muckenhoupt Weights

Abstract

The Hardy-Lorentz spaces for B-valued martingales space are stablished. We study the atomic decompositions, duality of the spaces .the interpolations and John-Nirenberg inequalities of martingale Hardy-Lorentz-Karamata spaces. The critical and the equivalence between pointwise Hardy inequalities presented. We deal with the characterization of the Triebel-Lizorkin spaces and weighted estimates for Littlewood - Paley functions with radial multipliers and bounds of singular integrals and discrete Littlewood-Palev analysis the realized Hardy inequalities under non-convexity measures ,for functions vanishing on boundary on the solid torus estimates for Littlewood-Paley Stein square functions. Calderon-Zygmund operators and limited ranges of Muckenhoupt weights are investigated.