Qp Spaces and Dirichlet Type Spaces. Issue 4 (1st December 2017)

• Record Type: Journal Article
• Authors: Bao, Guanlong; Gökhan Gögüs, Nihat; Pouliasis, Stamatis
• Is Part Of: Canadian mathematical bulletin =. Volume 60:Issue 4(2017)
• Page Start: 690
• Page End: 704
• Publication Date: 2017-12-01
• Subjects: 30H25 -- 31C25 -- 46E15; Qp space -- Dirichlet type space -- Möbius invariant function space; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/CMB-2017-006-1
• Languages: English
• ISSNs: 0008-4395

Abstract

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.