Carleson measures, Volterra integral operators and multipliers for $Q_K$ spaces

We characterize the positive Borel measures $\mu$ on the unit disc $\mathbb{D}$ for which the Möbius-invariant space $Q_K$ embeds continuously or compactly into $L^2(\mu )$. The characterization is given in terms of a discrete dyadic capacity $C_{K,\mathcal{R}}^{(b)}(\mu )$ built from a polar dyadic resolution of $\mathbb{D}$, and of an equivalent capacity $D_{K,\mathcal{R}}^{(b)}(\mu )$ 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).