Hilbert矩阵算子在Bloch型空间上的范数
Norm of the Hilbert Matrix Operator on Bloch-Type Spaces
中文总结 AI 辅助
本文确定了Hilbert矩阵算子在Bloch型空间上的精确范数，给出了显式公式和最大化公式，并完全刻画了达到范数的函数。
中文摘要 AI 辅助
我们确定了Hilbert矩阵算子$\mathcal H$在$\alpha$-Bloch空间$\mathcal B^α$上对于$1<α<2$的精确范数，以及所有范数为1的达到范数的函数。对于$1<α\leq3/2$，我们得到了用Gamma函数表示的精确范数的显式公式。对于$3/2<α<2$，我们得到了一个精确的一维最大化公式，并证明了相应的最大值在$(0,1)$的内点处取得。我们还确定了$\mathcal H:\mathcal B^α\to\mathcal B^α_{\log}$的精确范数。范数公式在临界参数$α=4/3$处发生变化，并且对于整个范围$1<α<2$都获得了精确范数。对于两个算子，达到范数的函数都被完全刻画。
英文摘要
We determine the exact norm of the Hilbert matrix operator $\mathcal H$ on the $α$-Bloch space $\mathcal B^α$ for $1<α<2$, together with all norm-attaining functions of norm one. For $1<α\leq3/2$, we obtain an explicit formula for the exact norm in terms of the Gamma function. For $3/2<α<2$, we obtain an exact one-dimensional maximization formula and show that the corresponding maximum is attained at an interior point of $(0,1)$. We also determine the exact norm of $\mathcal H:\mathcal B^α\to\mathcal B^α_{\log}$. The norm formula changes at the critical parameter $α=4/3$, and the exact norm is obtained for the full range $1<α<2$. The norm-attaining functions are completely characterized for both operators.
