  Let $A$ be a general expansive matrix and $X$ be a ball quasi-Banach function
space on $\mathbb R^n$, whose certain power (namely its convexification)
supports a Fefferman--Stein vector-valued maximal inequality and the associate
space of whose other power supports the boundedness of the powered
Hardy--Littlewood maximal operator. Let $H_X^A(\mathbb{R}^n)$ be the
anisotropic Hardy space associated with $A$ and $X$. The authors first prove
that the Fourier transform of $f\in H^A_{X}(\mathbb{R}^n)$ coincides with a
continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered
distributions. Moreover, the authors obtain a pointwise inequality that the
function $F$ 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 expansive
matrix $A$. Applying this, the authors further induce a higher order
convergence for the function $F$ at the origin and give a variant of the
Hardy--Littlewood inequality in $H^A_{X}(\mathbb{R}^n)$. All these results have
a wide range of applications. Particularly, the authors apply these results,
respectively, to classical (variable and mixed-norm) Lebesgue spaces, Morrey
spaces, Lorentz spaces, Orlicz spaces, Orlicz-slice spaces, and local
generalized Herz spaces and, even on the last five function spaces, the
obtained results are completely new.

Functional Analysis;Analysis of PDEs;Classical Analysis and ODEs