Hilbert Matrix and Its Norm on Weighted Bergman Spaces

Abstract

It is well known that the Hilbert matrix $${\mathrm {H}}$$ is bounded on weighted Bergman spaces $$A^p_\alpha $$ if and only if $$1<\alpha +2<p$$ with the conjectured norm $$\pi /\sin \frac{(\alpha +2)\pi }{p}$$ . The conjecture was confirmed in the case when $$\alpha =0$$ and also in the case when $$\alpha >0$$ and $$p\ge 2(\alpha +2)$$ , which reduces the conjecture in the case when $$\alpha >0$$ to the interval $$\alpha +2<p<2(\alpha +2)$$ . In the remaining case when $$-1<\alpha <0$$ and $$p>\alpha +2$$ there has been no progress so far in proving the conjecture, moreover, there is no even an explicit upper bound for the norm of the Hilbert matrix $${\mathrm {H}}$$ on weighted Bergman spaces $$A^p_\alpha $$ . In this paper we obtain results which are better than known related to the validity of the mentioned conjecture in the case when $$\alpha >0$$ and $$\alpha +2<p<2(\alpha +2)$$ . On the other hand, we also provide for the first time an explicit upper bound for the norm of the Hilbert matrix $${\mathrm {H}}$$ on weighted Bergman spaces $$A^p_\alpha $$ in the case when $$-1<\alpha <0$$ and $$p>\alpha +2$$ .