Hardy Spaces Associated with Inhomogeneous Generalized Herz Spaces

Abstract

The target of this chapter is devoted to establishing a complete real-variable theory of Hardy spaces associated with inhomogeneous generalized Herz spaces. Precisely, based on the inhomogeneous generalized Herz spaces studied in Chap. 7 , we first introduce inhomogeneous generalized Herz–Hardy spaces, inhomogeneous localized generalized Herz–Hardy spaces, and inhomogeneous weak generalized Herz–Hardy spaces in this chapter. Then, via using the known results about Hardy spaces $$H_X(\mathbb {R}^{n})$$ associated with ball quasi-Banach function spaces X as well as some improved characterizations of $$H_X(\mathbb {R}^{n})$$ established in ***Chaps. 4 through 6 , we investigate both the real-variable characterizations and their applications of these Hardy spaces associated with inhomogeneous generalized Herz spaces, which include various maximal function, the atomic, the molecular, and the Littlewood–Paley function characterizations as well as the boundedness of Calderón–Zygmund operators or pseudo-differential operators, duality, the properties of the Fourier transform, and the real interpolation theorems about these Herz–Hardy spaces.