Norm of the Hilbert matrix on Bergman spaces.

It is well known that the Hilbert matrix operator H is a bounded operator from the Bergman space A p into A p if and only if 2 < p < ∞ . In [5] it was shown that the norm of the Hilbert matrix operator H on the Bergman space A p is equal to π sin ⁡ 2 π p , when 4 ≤ p < ∞ , and it was also conjectured that ‖ H ‖ A p → A p = π sin ⁡ 2 π p , when 2 < p < 4 . In this paper we prove this conjecture. [ABSTRACT FROM AUTHOR]