Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces

Let X be a ball quasi-Banach function space satisfying some mild additional assumptions andH(X)(Double-struck capital R-n) the associated Hardy-type space. In this article, we first establish the finite atomic characterization ofH(x)(Double-struck capital R-n). As an application, we prove that the dual space ofH(X)(Double-struck capital R-n) is the Campanato space associated withX. For any given alpha is an element of (0, 1] and s is an element of DOUBLE-STRUCK CAPITAL Z(+), using the atomic and the Littlewood-Paley function characterizations ofH(X)(DOUBLE-STRUCK CAPITAL Z(n)), we also establish itss-order intrinsic square function characterizations, respectively, in terms of the intrinsic Lusin-area functionS(alpha,s), the intrinsicg-functiong(alpha,s), and the intrinsicg*(lambda)-functiong*(lambda alpha s), where lambda coincides with the best known range.