Characterization of $B_{p(\cdot)}$ Weight and Some Maximal Characterizations of Anisotropic Weighted Hardy–Lorentz Spaces with Variable Exponent

In the present paper the variable exponent anisotropic weighted Hardy-Lorentz spaces is introduced. We prove a characterization of a modular inequality of the classical Hardy operator on the decreasing cone of the variable exponent Lebesgue spaces which leads to a criterion of the boundedness for the Hardy–Littlewood operator on the variable exponent weighted Lorentz spaces. Furthermore, we get some characterizations of the variable exponent anisotropic weighted Hardy-Lorentz spaces by maximal operators. Also the completeness of these spaces are investigated. Specifying the weights and exponents we recover the existing results as well as we obtain new results in the new and old settings.