AI 中文总结
研究加权伯格曼空间上希尔伯特矩阵范数，针对偶数指数\(p = 2m\)给出范数公式成立条件，同时通过固定函数\(f_0(z)\)给出反例，证明贝塔函数范数公式并非对所有可允许参数成立，特别是对\(m \geq 550000\)的偶数指数\(p = 2m\)不成立。
AI 中文摘要
设\(A_\alpha^p\)为单位圆盘上的加权伯格曼空间，其中\(\alpha > -1\)。对于\(f(z)=\sum_{k=0}^{\infty}a_k z^k\in A_\alpha^p\)，考虑希尔伯特矩阵算子\(\mathcal{H}f(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}\frac{a_k}{n+k+1}\right)z^n =\int_0^1\frac{f(t)}{1-tz}\,dt\)。对于偶数指数\(p = 2m\)，当\(0 < a \leq m/(2m - 1)\)且\(a = (\alpha + 2)/(2m)\)时，证明了\(\|\mathcal{H}\|_{A_\alpha^{2m}\to A_\alpha^{2m}} = B(a, 1 - a)\)。还表明贝塔函数范数公式并非对所有可允许参数都成立。通过固定函数\(f_0(z)=(1 - z^2)^{-4/5}\)给出反例，并基于\(p = 1100000\)时的严格区间估计及\(p\)的单调性得出结果。特别是对于\(m \geq 550000\)的偶数指数\(p = 2m\)该公式不成立。
英文摘要
Let $A_α^p$ be the weighted Bergman space on the unit disk, where $α>-1$. For $f(z)=\sum_{k=0}^{\infty}a_k z^k\in A_α^p$, consider the Hilbert matrix operator $\mathcal{H}f(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}\frac{a_k}{n+k+1}\right)z^n =\int_0^1\frac{f(t)}{1-tz}\,dt$. For even exponents $p=2m$, we prove that $\|\mathcal{H}\|_{A_α^{2m}\to A_α^{2m}}=B(a,1-a)$, where $a=(α+2)/(2m)$, whenever $0<a\leq m/(2m-1)$. For $p=2,4,6,8,10$, the same formula holds throughout the admissible range. We also show that the beta-function norm formula does not hold for all admissible parameters. Set $a_0=800001/1000000$ and $α_p=a_0p-2$. Then, for every real $p\geq 1100000$, $\|\mathcal{H}\|_{A_{α_p}^p\to A_{α_p}^p}>B(a_0,1-a_0)$. The counterexample is based on the fixed function $f_0(z)=(1-z^2)^{-4/5}=\sum_{k=0}^{\infty}\frac{(4/5)_k}{k!}z^{2k}$. A rigorous interval estimate at $p=1100000$, together with monotonicity in $p$, yields the result on the entire half-line. In particular, the formula fails for every even exponent $p=2m$ with $m\geq 550000$.
Comments22 pages. We disprove Karapetrović's conjectured beta-function formula for the norm of the Hilbert matrix operator on weighted Bergman spaces
