Abstract view
$\mathcal{Q}_p$ 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
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 nonHilbert 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.