Embedding Theorems for Dirichlet Type Spaces

2019 ◽  
Vol 63 (1) ◽  
pp. 106-117 ◽  
Author(s):  
Songxiao Li ◽  
Junming Liu ◽  
Cheng Yuan
Keyword(s):  
Carleson Measure ◽  
Bergman Spaces ◽  
Embedding Theorem ◽  
Weighted Bergman Spaces ◽  
Borel Measures ◽  
Dirichlet Type ◽  
Volterra Operators ◽  
Type Spaces ◽  
Dirichlet Type Spaces ◽  
Analytic Function Spaces

AbstractWe use the Carleson measure-embedding theorem for weighted Bergman spaces to characterize the positive Borel measures $\unicode[STIX]{x1D707}$ on the unit disc such that certain analytic function spaces of Dirichlet type are embedded (compactly embedded) in certain tent spaces associated with a measure $\unicode[STIX]{x1D707}$. We apply these results to study Volterra operators and multipliers acting on the mentioned spaces of Dirichlet type.

scholarly journals Multiplicative Isometries and Isometric Zero-Divisors

2010 ◽  
Vol 62 (5) ◽  
pp. 961-974 ◽  
Author(s):  
Alexandru Aleman ◽  
Peter Duren ◽  
María J. Martín ◽  
Dragan Vukotić
Keyword(s):  
Banach Spaces ◽  
Bloch Space ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Zero Divisors ◽  
Pointwise Multipliers ◽  
Type Spaces ◽  
Dirichlet Type Spaces ◽  
Coefficient Multipliers ◽  
Spaces Of Analytic Functions

AbstractFor some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated.

QpSpaces and Dirichlet Type Spaces

2017 ◽  
Vol 60 (4) ◽  
pp. 690-704 ◽  
Author(s):  
Guanlong Bao ◽  
Nihat Gökhan Gögüs ◽  
Stamatis Pouliasis
Keyword(s):  
Hilbert Spaces ◽  
Invariant Function ◽  
Open Unit ◽  
Sufficient Condition ◽  
Borel Measures ◽  
Dirichlet Type ◽  
Type Spaces ◽  
Dirichlet Type Spaces ◽  
Decomposition Theorems ◽  
Dirichlet Type Space

AbstractIn this paper, we show that the Möbius invariant function space Qpcan be generated by variant Dirichlet type spaces 𝒟μ,pinduced by finite positive Borel measures μ on the open unit disk. A criterion for the equality between the space 𝒟μ,pand the usual Dirichlet type space 𝒟pis given. We obtain a sufficient condition to construct different 𝒟μ,pspaces and provide examples. We establish decomposition theorems for 𝒟μ,pspaces and prove that the non-Hilbert space Qpis equal to the intersection of Hilbert spaces 𝒟μ,p. As an application of the relation between Qpand 𝒟μ,pspaces, we also obtain that there exist different 𝒟μ,pspaces; this is a trick to prove the existence without constructing examples.

𝒩(p, q, s)-type spaces in the unit ball of ℂn(II): Carleson measure and its application

2020 ◽  
Vol 32 (1) ◽  
pp. 79-94 ◽  
Author(s):  
Bingyang Hu ◽  
Songxiao Li
Keyword(s):  
Unit Ball ◽  
Carleson Measure ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Weighted Hardy Spaces ◽  
Hadamard Products ◽  
New Class ◽  
Random Power Series ◽  
Type Spaces ◽  
Formula Type

AbstractThe purpose of this paper is to study a new class of function spaces, called {\mathcal{N}(p,q,s)}-type spaces, in the unit ball {{\mathbb{B}}} of {{\mathbb{C}}^{n}}. The Carleson measure on such spaces is investigated. Some embedding theorems among {\mathcal{N}(p,q,s)}-type spaces, weighted Bergman spaces and weighted Hardy spaces are established. As for applications, the Hadamard products and random power series on {\mathcal{N}(p,q,s)}-type spaces are also studied.

Embedding of Dirichlet type spaces into tent spaces and Volterra operators

2020 ◽  
pp. 1-12
Author(s):  
Ruishen Qian ◽  
Xiangling Zhu
Keyword(s):  
Integral Operators ◽  
Essential Norm ◽  
Function Spaces ◽  
General Function ◽  
Tent Spaces ◽  
Dirichlet Type ◽  
Volterra Operators ◽  
Type Spaces ◽  
Inclusion Mapping ◽  
Dirichlet Type Spaces

Abstract In this paper, we study the boundedness and compactness of the inclusion mapping from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to tent spaces. Meanwhile, the boundedness, compactness, and essential norm of Volterra integral operators from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to general function spaces are also investigated.

Products of Volterra type operators and composition operators between weighted Bergman spaces of infinite order and weighted Bloch type spaces

2010 ◽  
Vol 17 (3) ◽  
pp. 621-627 ◽  
Author(s):  
Elke Wolf
Keyword(s):  
Infinite Order ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Volterra Type ◽  
Volterra Operators ◽  
Type Spaces

Abstract We study the boundedness and compactness of products of Volterra operators and composition operators acting between weighted Bergman spaces of infinite order and weighted Bloch type spaces Bw .

scholarly journals Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Changbao Pang ◽  
Antti Perälä ◽  
Maofa Wang
Keyword(s):  
Borel Measure ◽  
Bergman Spaces ◽  
Embedding Theorem ◽  
Weighted Bergman Spaces ◽  
Embedding Theorems ◽  
Positive Borel Measure ◽  
Doubling Property ◽  
Area Operator ◽  
Special Cases ◽  
Upper Half Plane

AbstractWe establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$ d ω ( y ) d x with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$ ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

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