Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type
Let \(1< p<\infty\) and \(g\in L^\infty(0,2\pi)\). The authors prove that the Hankel operator \(H_g\) acts continuously on the Hardy space \(H^p\) of the unit disk, with \[ \| H_g\|_{H^p\to H^p}\leq {\| g\|_\infty\over \sin(\pi/p)}. \] In the particular case \(g(t)= ie^{-it}(\pi- t)\) \((0\leq t<2\pi)\), the Fourier coefficients of \(g\) are \(\widehat g(n)= 1/(n+ 1)\) for \(n\geq 0\), and therefore \(H_g:= H\) is the action of the Hilbert matrix on \(H^p\): \[ Hf(z)= \sum^\infty_{n=0} \Biggl(\sum^\infty_{k=0} {\widehat f(k)\over n+ k+ 1}\Biggr) z^n,\quad f\in H^p,\quad z\in\mathbb{D}. \] In this case, it is proved that \(\| H\|_{H^p\to H^p}= {\pi\over \sin(\pi/p)}\). The paper also explores the action of \(H\) on the Bergman spaces \(A^p\) for \(p> 2\) and the authors prove that \[ {\pi\over \sin(2\pi/p)}\leq\| H\|_{A^p\to A^p}\leq C{\pi\over \sin(2\pi/p)}, \] where the first inequality holds for \(2< p<\infty\) and the second for \(2< p< 4\), with \(C> 0\) an absolute constant.
scientific article; zbMATH DE number 5285677
scientific article