QpSpaces and Dirichlet Type Spaces

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.

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Characterizations of a Class of Dirichlet-Type Spaces and Related Operators

Journal of Mathematics ◽

10.1155/2020/6631047 ◽

2020 ◽

Vol 2020 ◽

pp. 1-6

Author(s):

Xiaosong Liu ◽

Songxiao Li

Keyword(s):

Boundary Value ◽

Type Space ◽

Dirichlet Type ◽

Type Spaces ◽

Dirichlet Type Spaces ◽

Dirichlet Type Space

In this paper, some characterizations are given in terms of the boundary value and Poisson extension for the Dirichlet-type space D μ . The multipliers of D μ and Hankel-type operators from D μ to L 2 P μ d A are also investigated.

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Higher order Hilbert-Schmidt Hankel forms and tensors of analytical kernels

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-14948 ◽

2005 ◽

Vol 96 (1) ◽

pp. 117 ◽

Cited By ~ 4

Author(s):

Sarah H. Ferguson ◽

Richard Rochberg

Keyword(s):

Hilbert Spaces ◽

Reproducing Kernel ◽

Higher Order ◽

Order Ideal ◽

Dirichlet Type ◽

Order Case ◽

Type Spaces ◽

Kernel Hilbert Spaces ◽

Dirichlet Type Spaces ◽

The Ideal

The symbols of $n^{\hbox{th}}$-order Hankel forms defined on the product of certain reproducing kernel Hilbert spaces $H(k_{i})$, $i=1,2$, in the Hilbert-Schmidt class are shown to coincide with the orthogonal complement in $H(k_{1})\otimes H(k_{2})$ of the ideal of polynomials which vanish up to order $n$ along the diagonal. For tensor products of weighted Bergman and Dirichlet type spaces (including the Hardy space) we introduce a higher order restriction map which allows us to identify the relative quotient of the $n^{\hbox{th}}$-order ideal modulo the $(n+1)^{\hbox{st}}$-order one as a direct sum of single variable Bergman and Dirichlet-type spaces. This generalizes the well understood $0^{\hbox{th}}$-order case.

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Embedding Theorems for Dirichlet Type Spaces

Canadian Mathematical Bulletin ◽

10.4153/s0008439519000201 ◽

2019 ◽

Vol 63 (1) ◽

pp. 106-117 ◽

Cited By ~ 1

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.

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