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

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.