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Let $A$ be a general expansive matrix and $X$ be a ball quasi-Banach functionspace on $\mathbb R^n$, whose certain power (namely its convexification)supports a Fefferman--Stein vector-valued maximal inequality and the associatespace of whose other power supports the boundedness of the poweredHardy--Littlewood maximal operator. Let $H_X^A(\mathbb{R}^n)$ be theanisotropic Hardy space associated with $A$ and $X$. The authors first provethat the Fourier transform of $f\in H^A_{X}(\mathbb{R}^n)$ coincides with acontinuous function $F$ on $\mathbb{R}^n$ in the sense of tempereddistributions. Moreover, the authors obtain a pointwise inequality that thefunction $F$ is less than the product of the anisotropic Hardy space norm of$f$ and a step function with respect to the transpose matrix of the expansivematrix $A$. Applying this, the authors further induce a higher orderconvergence for the function $F$ at the origin and give a variant of theHardy--Littlewood inequality in $H^A_{X}(\mathbb{R}^n)$. All these results havea wide range of applications. Particularly, the authors apply these results,respectively, to classical (variable and mixed-norm) Lebesgue spaces, Morreyspaces, Lorentz spaces, Orlicz spaces, Orlicz-slice spaces, and localgeneralized Herz spaces and, even on the last five function spaces, theobtained results are completely new.

Fourier Transform of Anisotropic Hardy Spaces Associated with Ball  Quasi-Banach Function Spaces and Its Applications to Hardy--Littlewood  Inequalities