On Hardy-type inequalities and Hankel transforms
If \(0<p\leq 1\) and \(H^p(\mathbb{R})\) is the real Hardy space, then \(f\in H^p(\mathbb{R})\) satisfies the Hardy-type inequality \[ \int_{\mathbb{R}}|\widehat f(\xi)|^p|\xi|^{p- 2}d\xi\leq C\| f\|^p_{H^p(\mathbb{R})}, \] where \(\widehat f\) stands for the Fourier transform of \(f\). The aim of the paper under review is to establish such an inequality for Hankel transforms \[ (H_\nu f)(y)= \int^\infty_0 f(x)J_\nu(xy) \sqrt{xy} dx,\quad y>0,\tag{\(*\)} \] where \(J_\nu(z)\) is the Bessel function of the first kind of the order \(\nu\). The formula \((*)\) is meaningful for \(f\in H^1(\mathbb{R})\) with \(\text{supp }f\subset [0,\infty)\). To define Hankel transforms for \(f\in H^p(\mathbb{R})\), \(0<p<1\), and \(\text{supp }f\subset [0,\infty)\), the author uses the fact that an element of the Lipschitz space \(\Lambda_{1/p-1}(\mathbb{R})\) represents a continuous linear functional on \(H^p(\mathbb{R})\).
scientific article; zbMATH DE number 1308339