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

Abstract

Let X be a ball quasi-Banach function space on ${\mathbb{R}}^n$ and $H_X\left({\mathbb{R}}^n\right)$ 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 $f\in H_X\left({\mathbb{R}}^n\right)$ coincides with a continuous function g on ${\mathbb{R}}^n$ 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.

Keywords:

• Ball quasi-Banach function space
• Hardy space
• Fourier transform
• Hardy–Littlewood inequality

2020 Mathematics Subject Classifications:

• Primary 42B35
• Secondary 42B30
• 42B10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Long Huang and Dachun Yang are supported by the National Natural Science Foundation of China [grant numbers 11971058 and 12071197] and the National Key Research and Development Program of China [grant number 2020YFA0712900], and Der-Chen Chang is partially supported by an NSF [grant number DMS-1408839] and a McDevitt Endowment Fund at Georgetown University.