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 \(p>\alpha +2\).

众所周知，希尔伯特矩阵\（{\ mathrm {H}} \）在且仅当\（1 <\ alpha +2 <p \）时，在加权Bergman空间\（A ^ p_ \ alpha \）上有界与猜想范数\ {\ pi / \ sin \ frac {（\ alpha +2）\ pi} {p} \）。在\（\ alpha = 0 \）的情况下以及在\（\ alpha> 0 \）和\（p \ ge 2（\ alpha +2）\）的情况下都证实了猜想，这减少了猜想在\（\ alpha> 0 \）到间隔\（\ alpha +2 <p <2（\ alpha +2）\）的情况下。在其余情况下，当\（-1 <\ alpha <0 \）和\（p> \ alpha +2 \）迄今为止，在证明这一猜想方面没有任何进展，而且，在加权Bergman空间\（A ^ p_ \ ）上，希尔伯特矩阵\（{\ mathrm {H}} \）的范数甚至没有明确的上限alpha \）。在本文中，当\（\ alpha> 0 \）和\（\ alpha +2 <p <2（\ alpha +2）\）时，我们得出的结果与上述猜想的有效性相比要好得多。在另一方面，我们还提供首次明确的上限为希尔伯特矩阵的范数\（{\ mathrm {H}} \）加权Bergman空间\（A ^ P_ \阿尔法\）的情况下当\（-1 <\ alpha <0 \）和\（p> \ alpha +2 \）时。