Fourier transform of variable anisotropic Hardy-Lorentz spaces and its applications

Abstract

Let $p(\cdot)$ be a measurable function on $\mathbb{R}^n$ satisfyingthe globally log-Hölder continuous condition.Its essential supremum $p_+$ and infimum $p_-$ satisfy$0<p_-\le~p_+\le1$.Let $H_A^{p(\cdot),q}(\mathbb{R}^n)$ be the variable anisotropicHardy-Lorentz spaces defined via the radial grand maximal function,where $q\in(0,1]$ and $A$ is an expansive matrix.In this article, byusing the atomic decomposition of $H_A^{p(\cdot),q}(\mathbb{R}^n)$,we prove that the Fourier transform of $f\in~H_A^{p(\cdot),q}(\mathbb{R}^n)$equals a continuous function $F$ on $\mathbb{R}^n$in the sense of tempered distributions.Moreover, the function $F$ can be pointwisely controlled bythe product of the $H_A^{p(\cdot),q}(\mathbb{R}^n)$ norm of $f$and the homogeneous quasi-norm associated with the transpose matrix of $A$.As applications, we 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 $H_A^{p(\cdot),q}(\mathbb{R}^n)$. All theseresults are new even for the isotropic Hardy-Lorentz spaces$H^{p,q}(\mathbb{R}^n)$ and they generalize the corresponding conclusionsof Taibleson and Weiss on classical Hardy spaces $H^{p}(\mathbb{R}^n)$.