The $p$--Bloch--Carleson measure problem for $0<p<\infty$: A complete characterization via Carleson tents

Abstract

We characterize the boundedness and compactness of the embedding $$\mathrm{id}:\mathcal{B}⟶L^p(\mu ),0<p<\infty ,$$ where $\mathcal{B}$ denotes the Bloch space and $\mu$ is a positive Borel measure on the unit disc. The criteria depend only on the dyadic Carleson tents. These results refine our earlier $L^2(\mu )$ characterization based on a full dyadic resolution of the disc and extend it to all positive exponents. The proof combines discretization on Carleson tents, representations of Bloch functions by series of analytic polynomials, and weighted matrix estimates. We also obtain corresponding boundedness and compactness characterizations for $$\text{id}:{\mathcal{Q}}_p⟶L^q(\mu ),0<p\le 1,0<q<\infty .$$ These results resolve the general $p$--Bloch--Carleson measure problem studied by Girela, Peláez, Pérez-González and Rättyä in 2008.

arXiv comment46 pages. Substantially revised and extended; title changed. The characterization based on a full dyadic resolution of the unit disc is refined to one involving only the dyadic Carleson tents, and extended from $L^2(\mu )$ to $L^p(\mu )$ for all $0<p<\infty$. Corresponding characterizations for ${\mathcal{Q}}_p$ spaces are also included. Comments welcome!