Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood-Paley Characterizations and Real Interpolation

Let X be a ball quasi-Banach function space on R-n. In this article, assuming that the powered Hardy-Littlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space WX and the associated space, the authors establish various Littlewood-Paley function characterizations of W H-X (R-n) under some weak assumptions on the Littlewood-Paley functions. The authors also prove that the real interpolation intermediate space (H-X(R-n), L-infinity (R-n))(theta,infinity) between the Hardy space associated with X, H-X(R-n), and the Lebesgue space L-infinity(R-n), is WHX1/(1-theta) (R-n), where theta is an element of. (0, 1). All these results are of wide applications. Particularly, when X := M-q(p) (R-n) (the Morrey space), X := L ((p)) over right arrow (R-n) (the mixed-norm Lebesgue space) and X := (E-Phi(q))(t) (R-n) (the Orlicz-slice space), all these results are even new; when X := L-omega(Phi) (Rn) (the weighted Orlicz space), the result on the real interpolation is new and, when X := Lp((center dot))(Rn) (the variable Lebesgue space) and X := L-omega(Phi) (R-n), the Littlewood-Paley function characterizations of WHX (R-n) obtained in this article improves the existing results via weakening the assumptions on the Littlewood-Paley functions.