On the holomorphic differential operator $\frac{d}{dz}: Q_K\to L^q(WdA)$

Abstract

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.