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

Abstract

We study embeddings and operators on the Möbius-invariant spaces $Q_K$ using two dyadic capacities. One capacity is defined by analytic packets; the other is defined by domination of positive semidefinite matrices. We prove that they are equivalent. They give criteria for bounded and compact embeddings $Q_K\to L^2(\mu )$, bounded and compact Volterra operators, and pointwise multipliers. For the weights $K(t)=t^p/{\log }^{\beta }(e/t)$, $0<p<1$ and $\beta \ge 1$, we prove $\mathcal{M}(Q_K)⊊H^{\infty }\cap Q_K^{\log }.$ This disproves the logarithmic multiplier conjecture recorded in Bao and Wulan's 2021 survey.