AI 中文总结
研究在特定假设下将哈代空间的刻画推广到哈代型空间\(X_A\)，并得出复合算子\(C_\varphi: \mathcal{B}^\omega \to X_A\)有界性和紧性的精确准则，推广了一维相关结果。
AI 中文摘要
已知哈代空间\(\mathrm{H}_{p}(\mathbb D)\)（\(0 < p < \infty\)）借助Littlewood-Paley g函数\(S f (\zeta) = \left(\int_0^1 |f' (r \zeta)|^2 (1 - r) \mathrm dr\right)^{1/2} \in \mathrm{L}_{p}\)进行刻画。在哈代 - 利特尔伍德极大算子\(M\)在\((X^\delta)'\)有界（\(\delta > 0\)）的假设下，此刻画被推广到单位圆\(\mathbb T\)上与拟巴拿赫格\(X\)对应的哈代型空间\(X_A\)。作为对复合算子\(C_\varphi\)的应用，得出\(C_\varphi: \mathcal{B}^\omega \to X_A\)有界性和紧性的精确准则，其中\(\mathcal{B}^\omega\)是具有对数凸径向权重\(\omega\)的加权布洛赫空间，推广了一维情形的近期结果。
英文摘要
The well-known characterization of Hardy spaces $\mathrm{H}_{p}(\mathbb D)$, $0 < p < \infty$, in terms of the Littlewood-Paley g-function $$ S f (ζ) = \left(\int_0^1 |f' (r ζ)|^2 (1 - r) \mathrm dr\right)^{1/2} \in \mathrm{L}_{p} $$ is generalized to Hardy-type spaces $X_A$ corresponding to quasi-Banach lattices $X$ on the unit circle $\mathbb T$ under the assumption that the Hardy-Littlewood maximal operator $M$ is bounded in $(X^δ)'$ with some $δ> 0$. As an application to composition operators $C_φ$, we derive an exact criterion for the boundedness and compactness of $C_φ: \mathcal{B}^ω\to X_A$, where $\mathcal{B}^ω= \{f \mid \sup |f'|/ω< \infty\}$ is the weighted Bloch space with a log-convex radial weight $ω$, generalizing recent results in the one-dimensional setting.
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