On the Hilbert matrix norm on positively indexed weighted Bergman spaces

Abstract

The Hilbert matrix is bounded on weighted Bergman spaces Aαp if and only if 1 < α+ 2 < p with the conjectured norm π/sin(α+2)πp . In the case of positively indexed weighted Bergman spaces, that is, in the case when α> 0 , the conjecture was confirmed for α≤ p , where α is a unique zero of the function Φα(x)=2x2-(4(α+2)+1)x+2α+2x+α+2 on the interval (α+ 2 , 2 (α+ 2)) . In this note we prove, that if α> 0 , then the conjecture is valid for all 3α4+2+(3α4+2)2-α+22≤p. This improves the best previously known result for all α>12 .