New Ball Campanato-Type Function Spaces and Their Applications

Let X be a ball quasi-Banach function space on R n . In this article, the authors first cleverly introduce some new ball Campanato-type function spaces associated with X . Then, under the additional assumptions that the Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued inequality on X and is also bounded on the associated space of X , the authors prove that these ball Campanato-type function spaces are the dual spaces of Hardy spaces associated with X . As applications, the authors establish several equivalent characterizations of these ball Campanato-type function spaces, which, together with the known atomic decomposition of tent spaces associated with X , further induces their Carleson measure characterizations. All these results have a wide range of applications. Particularly, even when these results are applied, respectively, to mixed-norm Hardy spaces, variable Hardy spaces, Orlicz–Hardy spaces, and Orlicz-slice Hardy spaces or the generalized amalgam Hardy spaces, the obtained conclusions also give their dual spaces for the full ranges which answer some open questions existing for some time.