Riesz Transform Characterization of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

Let X be a ball quasi-Banach function space satisfying some mild assumptions and H-X(R-n) the Hardy space associated with X. In this article, the authors introduce both the Hardy space H-X(R-+(n+1)) of harmonic functions and the Hardy space H-X(R-+(n+1)) of harmonic vectors, associated with X, and then establish the isomorphisms among H-X(R-n), H-X,H-2(R-+(n+1)), and HX,2(R-+(n+1)), where H-X,H-2(R+ (n+1)) and H-X,H-2(R-+(n+1) ) are, respectively, certain subspaces of H-X(R-+(n+1) ) and H-X (R-+(n+1)). Using these isomorphisms, the authors establish the first order Riesz transform characterization of HX(Rn). The higher order Riesz transform characterization of HX(Rn) is also obtained. The results obtained in this article have a wide range of generality and can be applied to classical Hardy spaces, weighted Hardy spaces, variable Hardy spaces, Herz-Hardy spaces, Lorentz-Hardy spaces, mixed-norm Hardy spaces, local generalized Herz-Hardy spaces, and mixed-norm Herz-Hardy spaces and all the obtained results on the aforementioned last five Hardy-type spaces are completely new.