Hilbert Matrix and Its Norm on Weighted Bergman Spaces

Abstract

It is well known that the Hilbert matrix H is bounded on weighted Bergman spaces Aαp if and only if 1 < α+ 2 < p with the conjectured norm π/sin(α+2)πp. The conjecture was confirmed in the case when α= 0 and also in the case when α> 0 and p≥ 2 (α+ 2) , which reduces the conjecture in the case when α> 0 to the interval α+ 2 < p< 2 (α+ 2). In the remaining case when - 1 < α< 0 and p> α+ 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 H on weighted Bergman spaces Aαp. In this paper we obtain results which are better than known related to the validity of the mentioned conjecture in the case when α> 0 and α+ 2 < p< 2 (α+ 2). On the other hand, we also provide for the first time an explicit upper bound for the norm of the Hilbert matrix H on weighted Bergman spaces Aαp in the case when - 1 < α< 0 and p> α+ 2.