Fourier transform of Hardy spaces associated with ball quasi-Banach function spaces*

Abstract

Let X be a ball quasi-Banach function space on and the associated Hardy space. In this article, under the assumptions that the Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued inequality on X and is bounded on the associated space of X as well as under a lower bound assumption on the X-quasi-norm of the characteristic function of balls, the authors show that the Fourier transform of coincides with a continuous function g on in the sense of tempered distributions and obtain a pointwise inequality about g and the Hardy space norm of f. Applying this, the authors further conclude a higher order convergence of the continuous function g at the origin and then give a variant of the Hardy–Littlewood inequality in the setting of Hardy spaces associated with X. All these results have a wide range of applications. Particularly, the authors apply these results, respectively, to mixed-norm Lebesgue spaces, variable Lebesgue spaces, and Orlicz spaces. Even in these special cases, the obtained results for variable Hardy spaces and Orlicz–Hardy spaces are totally new.