Hilbert matrix on Bergman spaces (Q1766875)

The Hilbert matrix \(H\) is defined as an infinite matrix with entries \(a_{i,j}=(i+j+1)^{-1}\), where \(i\), \(j\geq 0\). A classical inequality due to Hilbert allows us to determine the norm of \(H\) viewed as an operator acting on the standard sequence spaces \(l^{p}\). Since the space \(l^ 2\) can be identified with the classical Hardy space \(H^ 2\) and there is a similar way of viewing the Bergman space \(A^ 2\) of all square-integrable analytic functions on the unit disk in terms of its Taylor coefficients, this motivates analogous questions about the norm of \(H\) on the general Hardy spaces \(H^ p\) of the disk and on the Bergman spaces \(A^ p\) of analytic functions in the unit disk which are \(p\)-integrable with respect to area measure. The purpose of this paper is to study the norm of the operator \(H\) induced by the Hilbert matrix as an operator on \(A^{p}\). The action of the operator can be seen in the following way: it transforms the function whose sequence of Taylor coefficients is \((a_{n})_{n=0}^\infty\) into the function with Taylor coefficients \((\sum_{k=0}^\infty {a_{k}\over n+k+1})_{n=0}^\infty\). In the author's earlier paper with \textit{A. G. Siskakis} [Stud. Math. 140, No. 2, 191--198 (2000; Zbl 0980.47029)], an upper bound for the norm of this operator was obtained on the Hardy spaces \(H^{p}\), \(p>1\). The author shows that \(H\) is also bounded on \(A^ p\) when \(p>2\), but cannot possibly be bounded on \(A^ 2\) (using a duality argument). The question whether this operator is actually well defined on \(A^ 2\) is left open; it appears, though, that the answer is negative. The main result of the paper gives two upper bounds for the norm of \(H\) on \(A^ p\). The one obtained in the case \(4\leq p<\infty\) is analogous to the Hardy space result: \(\| H\| \leq \pi/\sin (2\pi/p)\), while a different one is obtained in the case \(2<p<4\). The method of proof relies on the representation of the Hilbert matrix in terms of a weighted composition operator: \[ H(f)(z) = \int_0^1 {1\over(t-1)z+1} f( {t\over(t-1)z+1} )dz. \] (Similar ideas have appeared earlier in several papers by Siskakis and coauthors for the Cesàro operator and other related operators.) The key point of the proof is the fact that when \(p\geq 4\), an estimate can be obtained that leads to a well-known identity for the Gamma-function. There are some technical difficulties when \(2<p<4\). Because of the way in which it is obtained, the upper bound for the norm in this case does not seem optimal. As the author points out, it is unclear what the best possible estimate is and also whether the estimate for \(p\geq 4\) is sharp. The author, thus, poses an interesting open question. He obtains a different estimate for the functions that vanish at the origin when \(2<p<4\). It may not be sharp either, but the idea of proof of Lemma 3 (used in obtaining the estimate) is nice and might be useful in proving other inequalities of this type.