Article contents
Qp Spaces and Dirichlet Type Spaces
Published online by Cambridge University Press: 20 November 2018
Abstract
In this paper, we show that the Möbius invariant function space ${{\mathcal{Q}}_{p}}$ can be generated by variant Dirichlet type spaces
${{\mathcal{D}}_{\mu ,p}}$ induced by finite positive Borel measures
$\mu $ on the open unit disk. A criterion for the equality between the space
${{\mathcal{D}}_{\mu ,p}}$ and the usual Dirichlet type space
${{\mathcal{D}}_{p}}$ is given. We obtain a sufficient condition to construct different
${{\mathcal{D}}_{\mu ,p}}$ spaces and provide examples. We establish decomposition theorems for
${{\mathcal{D}}_{\mu ,p}}$ spaces and prove that the non-Hilbert space
${{\mathcal{Q}}_{p}}$ is equal to the intersection of Hilbert spaces
${{\mathcal{D}}_{\mu ,p}}$. As an application of the relation between
${{\mathcal{Q}}_{p}}$ and
${{\mathcal{D}}_{\mu ,p}}$ spaces, we also obtain that there exist different
${{\mathcal{D}}_{\mu ,p}}$ spaces; this is a trick to prove the existence without constructing examples.
- Type
- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 2017
References
- 5
- Cited by