AI 中文总结
该研究利用二进容量测度刻画了全纯微分算子在$Q_K$空间到$L^q(WdA)$空间上的有界性与紧性，推广了$\mathcal{Q}_p$空间的相关结果，并解决了Zhao 2009年关于复合算子的公开问题。
AI 中文摘要
本文利用二进容量测度，得到了微分算子 $\frac{d}{dz}:Q_K\longrightarrow L^q(W\\,dA)$（$0<q<\infty$）有界性与紧性的非检验刻画。我们还刻画了 $q\to0^+$ 时的极限情形，采用对数几何均值形式表述；而端点 $q=\infty$ 则通过标准检验论证单独处理。这些结果将此前关于 $\mathcal{Q}_p$ 空间的研究工作大幅推广至 $Q_K$ 空间的一般框架。作为应用，我们刻画了不同 $Q_K$ 空间之间的复合算子与Volterra型积分算子。特别地，本文建立的非对角刻画，结合此前已得到的对角情形，完全解决了Zhao于2009年提出的关于 $\mathcal{Q}_p$ 空间之间复合算子的公开问题。
英文摘要
In this paper, we obtain non-testing characterizations, in terms of dyadic capacity gauges, of the boundedness and compactness of the differentiation operator $$ \frac{d}{dz}:Q_K\longrightarrow L^q(W\,dA), \qquad 0<q<\infty. $$ We also characterize the limiting case as $q\to0^+$, formulated in terms of a logarithmic geometric mean, while the endpoint $q=\infty$ is treated separately using a standard testing argument. These results greatly extend the previous work on ${\mathcal Q}_p$-spaces to the general setting of $Q_K$-spaces. As applications, we characterize composition operators and Volterra-type integral operators between different $Q_K$-spaces. In particular, the off-diagonal characterization established here, together with the previously established diagonal case, completely resolves Zhao's 2009 open question on composition operators between ${\mathcal Q}_p$-spaces.
