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Let $\vec{p}\in(0,1]^n$ be a $n$-dimensional vector and $A$ a dilation. Let$H_A^{\vec{p}}(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy spacedefined via the radial maximal function. Using the known atomiccharacterization of $H_{A}^{\vec{p}}(\mathbb{R}^n)$ and establishing a uniformestimate for corresponding atoms, the authors prove that the Fourier transformof $f\in H_A^{\vec{p}}(\mathbb{R}^n)$ coincides with a continuous function $F$on $\mathbb{R}^n$ in the sense of tempered distributions. Moreover, thefunction $F$ can be controlled pointwisely by the product of the Hardy spacenorm of $f$ and a step function with respect to the transpose matrix of $A$. Asapplications, the authors obtain a higher order of convergence for the function$F$ at the origin, and an analogue of Hardy--Littlewood inequalities in thepresent setting of $H_A^{\vec{p}}(\mathbb{R}^n)$.

Fourier Transform of Anisotropic Mixed-norm Hardy Spaces with Applications to Hardy-Littlewood Inequalities