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$\mathcal{Q}_p$ spaces and Dirichlet type spaces

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

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