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# $\mathcal{Q}_p$ Spaces and Dirichlet Type Spaces

Published:2017-03-09
Printed: Dec 2017
• Guanlong Bao,
Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China
• Nıhat Gökhan Göğüş,
Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, Istanbul 34956, Turkey
• Stamatis Pouliasis,
Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, Istanbul 34956, Turkey
 Format: LaTeX MathJax PDF

## 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 we 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.
 Keywords: $\mathcal {Q}_p$ space, Dirichlet type space, Möbius invariant function space
 MSC Classifications: 30H25 - Besov spaces and $Q_p$-spaces 31C25 - Dirichlet spaces 46E15 - Banach spaces of continuous, differentiable or analytic functions

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