Q<i><sub>p</sub></i>Spaces and Dirichlet Type Spaces

Bao, Guanlong; Göğüş, Nıhat Gökhan; Pouliasis, Stamatis

doi:10.4153/cmb-2017-006-1

Cited by 6 publications

(3 citation statements)

References 14 publications

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“…These weights were considered in [3,2] and were used there to study certain weighted Dirichlet spaces and the associated Möbius invariant function spaces.…”

Section: Examples Of Weightsmentioning

confidence: 99%

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A Hardy–Littlewood theorem for Bergman spaces

Bao¹,

Wulan²,

Zhu³

2018

Ann. Acad. Sci. Fenn. Math.

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We study positive weight functions ω(z) on the unit disk D such that D |f (z)| p ω(z) dA(z) < ∞ if and only ifˆD (1 − |z| 2) p |f ′ (z)| p ω(z) dA(z) < ∞, where f is analytic on D and dA is area measure on D. We obtain some conditions on ω that imply the equivalence above, and we apply our conditions to several important classes of weights that have appeared in the literature before. D |f (z) − f (0)| p dA α (z) ∼ˆD(1 − |z| 2) p |f ′ (z)| p dA α (z) for f ∈ H(D). See [6, 14]. It is then natural to ask for conditions on finite positive Borel measures µ on D such that D |f (z) − f (0)| p dµ(z) ∼ˆD(1 − |z| 2) p |f ′ (z)| p dµ(z)

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“…These weights were considered in [3,2] and were used there to study certain weighted Dirichlet spaces and the associated Möbius invariant function spaces.…”

Section: Examples Of Weightsmentioning

confidence: 99%

“…Thus our condition in Theorem B is fine enough to detect the cut-off point σ = 2 for the weights ω σ . Note that the weight functions ω σ (z) were used in [3,2] to study Dirichlet type spaces and their corresponding Möbius invariant counterparts.…”

Section: Introductionmentioning

confidence: 99%

A Hardy–Littlewood theorem for Bergman spaces

Bao¹,

Wulan²,

Zhu³

2018

Ann. Acad. Sci. Fenn. Math.

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“…Comparing with Q K and Q p spaces, M(D µ ) spaces connect D and BMOA more smoothly. We refer to [11] for a recent investigation of Q p spaces and a class of Dirichlet-type spaces D µ,p induced by finite positive Borel measures µ on D.…”

Section: The Equality Between Bmo a And Q K Via M(d µ ) Spacesmentioning

confidence: 99%

Möbius Invariant Function Spaces and Dirichlet Spaces With Superharmonic Weights

Bao¹,

Mashreghi²,

Pouliasis

et al. 2018

J. Aust. Math. Soc.

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Let ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ be Dirichlet spaces with superharmonic weights induced by positive Borel measures $\unicode[STIX]{x1D707}$ on the open unit disk. Denote by $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ Möbius invariant function spaces generated by ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$. In this paper, we investigate the relation among ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$, $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ and some Möbius invariant function spaces, such as the space $BMOA$ of analytic functions on the open unit disk with boundary values of bounded mean oscillation and the Dirichlet space. Applying the relation between $BMOA$ and $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$, under the assumption that the weight function $K$ is concave, we characterize the function $K$ such that ${\mathcal{Q}}_{K}=BMOA$. We also describe inner functions in $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ spaces.

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