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

Abstract

Let p(•) : R n → (0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition and A a general expansive matrix on R n . Let H p(•) A (R n ) be the variable anisotropic Hardy space associated with A defined via the radial maximal function. In this article, via the known atomic characterization of H p(•)A (R n ) and establishing two useful estimates on anisotropic variable atoms, the author shows that the Fourier transform f of f ∈ H p (•) A (R n ) coincides with a continuous function F in the sense of tempered distributions, and F satisfies a pointwise inequality which contains a step function with respect to A as well as the Hardy space norm of f . As applications, the author also obtains a higher order convergence of the continuous function F at the origin. Finally, an analogue of the Hardy-Littlewood inequality in the variable anisotropic Hardy space setting is also presented. All these results are new even in the classical isotropic setting.