关于连续半群的一致收敛性
On the uniform convergence of continuous semigroups
AI总结：
本文研究复平面区域上连续半群的一致收敛性，将单位圆盘上 Gumenyuk 的结果加强为最优速率 O(√t)，并利用无穷小生成元刻画右半平面上半群一致收敛到恒等映射的条件。
摘要：
设 $Ω$ 为复平面 $\mathbb C$ 中的一个区域，并设 $\{Φ_t \}_{t\ge 0}$ 是 $Ω$ 上的连续半群；即对每个 $t\ge 0$，$Φ_t:Ω\toΩ$ 是全纯的，$Φ_0(z)=z$ 对每个 $z\inΩ$ 成立，$Φ_t\circΦ_s=Φ_{s+t}$ 对每个 $s$、$t\ge 0$ 成立，并且当 $t\to0^+$ 时，$Φ_t(z)\to z$ 在 $Ω$ 的紧子集上一致成立。尽管该定义只要求在紧子集上一致收敛，P. Gumenyuk 在 2014 年证明，当 $Ω$ 是单位圆盘时，收敛在整个 $\mathbb D$ 上是一致的。本文加强了 Gumenyuk 的结果，证明对 $\mathbb D$ 上的每个连续半群 $\{Φ_t\}_{t\ge 0}$，当 $t\to0^+$ 时有 $$\sup_{z\in\mathbb D}|Φ_t(z)-z|= O(\sqrt t)。$$ 此外，我们给出一个例子，说明 $O(\sqrt t)$ 是对 $\mathbb D$ 上所有半群都成立的一致收敛速率的最优可能值。当 $Ω$ 是右半平面 $\mathbb C_+$ 时，我们考虑以 $\infty$ 为 Denjoy-Wolff 点的半群 $\{Φ_t\}$。不难证明 Gumenyuk 的结果对这些半群不再成立。我们的第二个结果用无穷小生成元刻画了这类连续半群何时在 $t\to0^+$ 时一致收敛到恒等映射。具体而言，该收敛成立当且仅当半群的无穷小生成元在半平面 $\{z\in \mathbb C:\\, \Re z>1\}$ 上有界。在这种情况下，我们还能证明收敛速率在 $t\to0^+$ 时同样为 $O(\sqrt{t})$。该结果的一个应用实例是当半群属于 Gordon-Hedenmalm 类（即在 Dirichlet 级数的 Hardy 空间上产生有界复合算子的那一类）时。这些结果证明中的一个重要成分是调和测度，我们通过 M. Lavrentiev 的一个经典结果加以处理。
英文摘要：
Let $Ω$ be a region in the complex plane $\mathbb C$ and let $\{Φ_t \}_{t\ge 0}$ be a continuous semigroup of functions on $Ω$; that is, $Φ_t:Ω\toΩ$ is holomorphic for every $t\ge 0$, $Φ_0(z)=z$, for every $z\inΩ$, $Φ_t\circΦ_s=Φ_{s+t}$, for every $s$, $t\ge 0$, and \[ Φ_t(z)\to z\,,\quad t\to0^+, \] uniformly on compact subsets of $Ω$. Despite this definition only requires the uniform convergence on compact subsets, P. Gumenyuk proved in 2014 that, when $Ω$ is the unit disc, the convergence is uniform on the whole $\mathbb D$. In this paper, we enhance Gumenyuk's result by proving that for every continuous semigroup $\{Φ_t\}_{t\ge 0}$ on $\mathbb D$ we have $$ \sup_{z\in\mathbb D}|Φ_t(z)-z|= O(\sqrt t), \ t\to0^+. $$ In addition, we provide an example showing that $O(\sqrt t)$ is the best possible rate of uniform convergence valid for all semigroups on $\mathbb D$. When $Ω$ is the right half-plane $\mathbb C_+$, we consider semigroups $\{Φ_t\}$ with $\infty$ as its Denjoy-Wolff point. It is not difficult to show that Gumenyuk's result is no longer true for these semigroups. Our second result characterises when such continuous semigroups converge uniformly to the identity, as $t\to0^+$, in terms of their infinitesimal generators. Namely, this convergence holds if and only if the infinitesimal generator of the semigroup is bounded in the half-plane $\{z\in \mathbb C:\, \Re z>1\}$. In this case, we can also prove that the rate of convergence is again $O(\sqrt{t})$, as $t\to0^+$. An example of application of this result is when the semigroup is in the Gordon-Hedenmalm class (the one producing bounded composition operators on Hardy spaces of Dirichlet series). An important ingredient in the proofs of these results is harmonic measure, which we have done through a classic result of M. Lavrentiev.
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