AI 中文总结
本文针对Hardy空间上广义Hilbert算子的有界性开放问题，证明当2<p<∞时，Λ(p,1/p)条件不蕴含算子有界，给出H^p→H^2等映射的精确刻画，还解决了缺项符号类的对应问题及算子紧性刻画。
AI 中文摘要
设g是单位圆盘上的解析函数，考虑广义Hilbert算子$$\n\n\n\n\n\n\n\n\n\ng'(tz)\n\ndt\n\n$$当1<p≤2时，$\n\ng$在平均Lipschitz条件$g∈Λ(p,1/p)$下可刻画$\n\ng$在$H^p$上的有界性，而当$2<p<∞$时该问题仍未解决。近期研究表明，当$2<p<∞$时，条件$g∈Λ(p,1/p)$并不蕴含$\n\ng$在$H^p$上的有界性[GuoTang2026]。本文证明，在$2<p<∞$的范围内，该条件远不充分：对每个$2<p<∞$，都存在一个函数$g∈Λ(p,1/p)$，使得$\n\ng$甚至不是从$H^p$到$H^1$的有界算子。核心工作是对所有$1≤p≤∞$，精确刻画$\n\ng:H^p→H^2$的有界性；特别地，当$2<p<∞$时，该映射有界当且仅当$g'$属于某个混合范数空间。对于缺项符号，相同的混合范数条件也可刻画$\n\ng$在$H^p$上的有界性，从而在该类符号内完全解决了这一开放问题。本文还证明，对$1≤q≤∞$，$\n\ng:H^1→H^q$的有界性由条件$g'∈H^q$刻画，并刻画了上述情形下$\n\ng$的紧性。
英文摘要
Let $g$ be analytic in the unit disc and consider the generalized Hilbert operator $$ \mathcal{H}_g(f)(z)=\int_0^1 f(t)g'(tz)\, dt. $$ The boundedness of $\mathcal H_g$ on $H^p$ is characterized by the mean Lipschitz condition $g\inΛ\left(p,\frac{1}{p}\right)$ when $1<p\leq2$, while the problem remains open for $2<p<\infty$. It has been recently proved that the condition $g\inΛ\left(p,\frac{1}{p}\right)$ does not imply the boundedness of $\mathcal H_g$ on $H^p$, $2<p<\infty$ \cite{GuoTang2026}. We show that this condition is far from sufficient in the latter range: for every $2<p<\infty$, there exists a function $g\inΛ\left(p,\frac{1}{p}\right)$ such that $\mathcal H_g$ is not bounded even from $H^p$ into $H^1$. The main ingredient is an exact characterization of the boundedness of $\mathcal{H}_g:H^p\to H^2$ for all $1\leq p\leq\infty$. In particular, when $2<p<\infty$, this mapping is bounded if and only if $g'$ belongs to a certain mixed-norm space. For lacunary symbols, the same mixed-norm condition also characterizes the boundedness of $\mathcal H_g$ on $H^p$, and hence gives a complete solution of the open problem within this class of symbols. We also show that, for $1\leq q\leq\infty$, boundedness of $\mathcal H_g:H^1\to H^q$ is characterized by the condition $g'\in H^q$. We also characterize compactness of $\mathcal H_g$ in the aforementioned cases.
