AI 中文总结
该研究刻画了Q_K空间到L^2(\u03bc)的嵌入性，建立了两种容量的等价性，解决了Volterra积分算子有界性问题，并回答了Q_K乘子的开放问题。
AI 中文摘要
我们刻画了单位圆盘$\n\b{D}$上的正Borel测度$\n\b{\u03bc}$，使得Möbius不变空间$Q_K$连续或紧嵌入到$L^2(\n\b{\u03bc})$中。该刻画由从$\n\b{D}$的极二进分解构造的离散二进容量$C^{(b)}_{K,\n\b{\u03c1}}(\n\b{\u03bc})$，以及表示为半定规划的等价容量$D^{(b)}_{K,\n\b{\u03c1}}(\n\b{\u03bc})$给出。通过锥对偶和复Grothendieck不等式建立了两种容量的等价性。作为应用，我们刻画了$Q_K$上Volterra积分算子$T_g$的有界性与紧性，填补并完全解决了Li和Wulan（2010）建立的充分条件与必要条件之间的缺口。我们还得到了$Q_K$上逐点乘子$\n\b{\u039c}(Q_K)$的完全非测试刻画，从而回答了Bao和Wulan（2021）综述中提出的一个开放问题。
英文摘要
We characterize the positive Borel measures $μ$ on the unit disc $\mathbb{D}$ for which the Möbius-invariant space $Q_K$ embeds continuously or compactly into $L^2(μ)$. The characterization is given in terms of a discrete dyadic capacity $C^{(b)}_{K,\mathcal{R}}(μ)$ built from a polar dyadic resolution of $\mathbb{D}$, and of an equivalent capacity $D^{(b)}_{K,\mathcal{R}}(μ)$ expressed as a semidefinite program. The equivalence of the two capacities is established through conic duality and the complex Grothendieck inequality. As an application, we characterize the boundedness and compactness of the Volterra integral operator $T_g$ on $Q_K$, bridging and completely resolving the gap between the sufficient and necessary conditions established by Li and Wulan (2010). We also obtain a complete non-testing characterization of the pointwise multipliers $\mathcal{M}(Q_K)$ on $Q_K$, thereby answering an open problem posed in the survey of Bao and Wulan (2021).
