Fourier transform of anisotropic Hardy spaces (Q2839300)

Estimates for the Fourier transform \(\hat{f}\) of functions belonging to the anisotropic Hardy space associated with a dilation matrix \(A\) are studied. The anisotropic Hardy space \(H_A^p(\mathbb{R}^n)\) consists of all tempered distributions \(f \in {\mathcal S}'\) so that the maximal radial function associated to anisotropic dilations of a function \(\varphi\) in the Schwartz class, \(\varphi_k(x)=|\det A|^k \varphi(A^k x)\), \(k\in \mathbb{Z}\), is in \(L^p(\mathbb{R}^n)\).NEWLINENEWLINEThe main result is the following: Let \(p\in (0,1]\). If \(f\in H_A^p(\mathbb{R}^n)\), then \(\hat{f}\) is a continuous function and satisfies NEWLINE\[NEWLINE |\hat{f}(\xi)|\leq C \|f\|_{H_A^p} \rho_{*}(\xi)^{1/p-1},NEWLINE\]NEWLINE where \(C=C(A,p)\) and \(\rho_{*}\) is the quasi-norm associated with the transposed matrix \(A^{*}\).NEWLINENEWLINEThe result leads to similar consequences as in the classical isotropic setting. In particular, it implies the necessity of vanishing moments for anisotropic atoms in \(H_A^p\). Also, conditions for a function \(m\) to be a multiplier on \(H_A^p\) are obtained. Lastly, as in the work of \textit{J. García-Cuerva} and \textit{V. I. Kolyada} [Math. Nachr. 228, 123--144 (2001; Zbl 1011.42007)], the use of rearrangement functions allows to obtain an improvement of the Hardy-Littlewood's inequality which implies the integrability of \( |\hat{f}(\xi)|^p \rho_{*}(\xi)^{p-2}\).