## What is/are Dirichlet Space?

Dirichlet Space - The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Peláez, who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces.^{[1]}This paper proves two-sided estimates for the Dirichlet heat kernel on inner uniform domains in metric measure Dirichlet spaces satisfying the volume doubling condition, the Poincare inequality, and a cutoff Sobolev inequality.

^{[2]}Then we prove that operator $$S_k\,(k\ge 1)$$ S k ( k ≥ 1 ) is a Hilbert–Schmidt operator on the Dirichlet space of $${\mathbb {D}}$$ D , but fails to be the Hilbert–Schmidt operator on the Dirichlet space of $${\mathbb {D}}^{2}$$ D 2.

^{[3]}Our analysis include the Hardy spaces, and suitable generalizations of the classical Bloch and Dirichlet spaces.

^{[4]}We give a characterization of onto interpolating sequences with finite associated measure for the Dirichlet space in terms of condenser capacity.

^{[5]}In the framework of a strictly local regular Dirichlet space ${\bf X}$ we introduce the subspaces $PW_{\omega},\>\>\omega>0,$ of Paley-Wiener functions of bandwidth $\omega$.

^{[6]}We characterize the reducing subspaces of S 2 ⁎ S 1 or S 1 S 2 ⁎ as wandering subspaces with additional structures, and give a unified way to describe the reducing subspaces of Toeplitz operators induced by non-analytic monomial on weighted Hardy spaces of several variables, which including many classical function spaces, such as weighted Bergman space and Dirichlet space over the polydisk.

^{[7]}A special case of the main result states the following: Let $n \geq 2$, and let $\mathcal{Q}_i$, $i = 1, \ldots , n$, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\mathbb {C}$.

^{[8]}We characterize pluriharmonic symbols of the finite rank little Hankel operators on the Dirichlet space of the unit polydisk.

^{[9]}We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology.

^{[10]}We show the validity of a complete description of closed ideals of the algebra which is a commutative Banach algebra , that endowed with a pointwise operations act on Dirichlet space of algebra of series of analytic functions on the unit disk satisfying the Lipscitz condition of order of square sequence obtained by (Brahim Bouya, 2008), we introduce and deal with approximation square functions which is an outer functions to produce and show results in.

^{[11]}In this paper, we discuss relationships between the continuous embeddings of Dirichlet spaces ( F , E 1 ) into Lebesgue spaces and the integrability of the associated resolvent kernel r α ( x , y ).

^{[12]}Zhao as part of the large family of F (p, q, s) spaces, and are the Möbius invariant subspaces of the Dirichlet spaces D p−1.

^{[13]}In this paper, we introduce a family of Dirichlet spaces $$\{{\mathcal {D}}_n\}_{n\in {\mathbb {N}}}$$ { D n } n ∈ N.

^{[14]}This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball.

^{[15]}The Dirichlet space follows as a corollary.

^{[16]}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.

^{[17]}Let D denote the classical Dirichlet space of analytic functions on the open unit disc whose derivative is square area integrable.

^{[18]}On the Dirichlet space of the unit ball, we study some algebraic properties of Toeplitz operators.

^{[19]}Let μ be a positive ûnite Borel measure on the unit circle andD(μ) the associated harmonically weighted Dirichlet space.

^{[20]}In this paper, we study the spectrum of Toeplitz operators on the Dirichlet space D with bounded conjugate analytic symbols and give a characterization of the kernels of Toeplitz operators with harmonic symbols in P ‾ + M ( D ).

^{[21]}The main objects under study are quotients of multiplier algebras of certain complete Nevanlinna--Pick spaces, examples of which include the Drury--Arveson space on the ball and the Dirichlet space on the disc.

^{[22]}In the former case, a formula of H\"ormander's is used to show that the target space can be replaced with the Dirichlet space, along with additional explicit terms.

^{[23]}We show that in a scale of weighted Dirichlet spaces $$D_{\alpha }$$ D α , including the Bergman space, given any finite Blaschke product B there exists an equivalent norm in $$D_{\alpha }$$ D α such that B satisfies the wandering subspace property with respect to such norm.

^{[24]}As a novel bridge between the Dirichlet space $${\mathcal {D}}$$D, the John–Nirenberg space $$\mathcal {BMOA}$$BMOA, and the Bloch space $${\mathcal {B}}$$B on the unit disk, the Moebius invariant analytic function space $$Q_{\log ,p}$$Qlog,p founded directly on a Moebius invariant isoperimetry is discovered in accordance with the Moebius invariant inclusion chain $${\mathcal {H}}^\infty \subsetneq \mathcal {BMOA}\subsetneq {\mathcal {B}}$$H∞⊊BMOA⊊B, where $${\mathcal {H}}^\infty $$H∞ is the Hardy algebra of all bounded analytic functions on the unit disk.

^{[25]}In this paper we characterize those power series that act as Hadamard multipliers on all weighted Dirichlet spaces on the disk with superharmonic weights, and we obtain sharp estimates on the corresponding multiplier norms.

^{[26]}In this article, methods from sub-Hardy Hilbert spaces such as the de Branges-Rovnyak spaces and local Dirichlet spaces are used to investigate Baez-Duarte's Hilbert space reformulation of the Riemann hypothesis (RH).

^{[27]}We study the capacity in the sense of Beurling–Deny associated with the Dirichlet space D ( μ ) where μ is a finite positive Borel measure on the unit circle.

^{[28]}In this paper, we discuss the relationship between the adjoints of linear fractional composition operators with respect to two equivalent norms on the Dirichlet space of the disk or the ball.

^{[29]}The Gaussian analytic function we study arises in connection with the classical Dirichlet space, which is naturally Mobius invariant.

^{[30]}Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $\Sigma$ by elements of Bergman space and Dirichlet space on fixed regions in $R$ containing $\Sigma$.

^{[31]}We characterize closed range composition operators on the Dirichlet space for a particular class of composition symbols.

^{[32]}In this paper, we completely characterize the order boundedness of weighted composition operators between different weighted Dirichlet spaces and different derivative Hardy spaces.

^{[33]}We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc.

^{[34]}Martinez-Avendano raised the open question that whether or not the weighted Dirichlet spaces D α p can support hypercyclic composition operators when p − 2 α p.

^{[35]}This study aims at rational approximation of a class of weighted Hardy spaces, including the classical Bergman space, the weighted Bergman spaces, the Hardy space, the Dirichlet space and the Hardy–Sobolev spaces.

^{[36]}ABSTRACT In this paper, the boundedness and compactness of the differences of generalized integration operators from the space of Cauchy integral transforms to the Bloch-type spaces and the weighted Dirichlet spaces are investigated.

^{[37]}

## Weighted Dirichlet Space

The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Peláez, who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces.^{[1]}Let μ be a positive ûnite Borel measure on the unit circle andD(μ) the associated harmonically weighted Dirichlet space.

^{[2]}We show that in a scale of weighted Dirichlet spaces $$D_{\alpha }$$ D α , including the Bergman space, given any finite Blaschke product B there exists an equivalent norm in $$D_{\alpha }$$ D α such that B satisfies the wandering subspace property with respect to such norm.

^{[3]}In this paper we characterize those power series that act as Hadamard multipliers on all weighted Dirichlet spaces on the disk with superharmonic weights, and we obtain sharp estimates on the corresponding multiplier norms.

^{[4]}In this paper, we completely characterize the order boundedness of weighted composition operators between different weighted Dirichlet spaces and different derivative Hardy spaces.

^{[5]}Martinez-Avendano raised the open question that whether or not the weighted Dirichlet spaces D α p can support hypercyclic composition operators when p − 2 α p.

^{[6]}ABSTRACT In this paper, the boundedness and compactness of the differences of generalized integration operators from the space of Cauchy integral transforms to the Bloch-type spaces and the weighted Dirichlet spaces are investigated.

^{[7]}

## Classical Dirichlet Space

This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball.^{[1]}Let D denote the classical Dirichlet space of analytic functions on the open unit disc whose derivative is square area integrable.

^{[2]}The Gaussian analytic function we study arises in connection with the classical Dirichlet space, which is naturally Mobius invariant.

^{[3]}

## Regular Dirichlet Space

In the framework of a strictly local regular Dirichlet space ${\bf X}$ we introduce the subspaces $PW_{\omega},\>\>\omega>0,$ of Paley-Wiener functions of bandwidth $\omega$.^{[1]}We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology.

^{[2]}

## dirichlet space d

In this paper, we study the spectrum of Toeplitz operators on the Dirichlet space D with bounded conjugate analytic symbols and give a characterization of the kernels of Toeplitz operators with harmonic symbols in P ‾ + M ( D ).^{[1]}We study the capacity in the sense of Beurling–Deny associated with the Dirichlet space D ( μ ) where μ is a finite positive Borel measure on the unit circle.

^{[2]}