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 ℝⁿ, 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 ^A_X (ℝⁿ) be the anisotropic Hardy space associated with A and X. The authors first prove that the Fourier transform of f ∈ H ^A_X (ℝⁿ) coincides with a continuous function F on ℝⁿ 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 (ℝⁿ). All these results have a wide range of applications. Particularly, the authors apply these results, respectively, to classical (variable and mixed-norm) Lebesgue spaces, Lorentz spaces, Orlicz spaces, Orlicz-slice spaces, and local generalized Herz spaces and, even on the last four function spaces, the obtained results are completely new.