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

Abstract

Let p ∈ (0, 1] n be a n-dimensional vector and A a dilation. Let H p A (R n ) denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of H p A (R n ) and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of f ∈ H p A (R n ) coincides with a continuous function F on R n in the sense of tempered distributions. Moreover, the function F can be controlled pointwisely by the product of the Hardy space norm of f and a step function with respect to the transpose matrix of A. As applications, the authors obtain a higher order of convergence for the function F at the origin, and an analogue of Hardy-Littlewood inequalities in the present setting of H p A (R n ).