Weighted Dirichlet Spaces Research Articles

Weighted Dirichlet spaces that are de Branges-Rovnyak spaces with equivalent norms

For a positive Borel measure μ in the unit disc, we examine which weighted Dirichlet spaces Dμ can be identified with de Branges-Rovnyak spaces Hb, with equivalent norms. We prove a necessary condition for the equality Dμ=Hb and we explore its consequences. For Carleson measures μ, we give a necessary and sufficient condition for the equality Dμ=Hbμ, for a certain outer function bμ related with the balayage of μ on the unit circle, and we provide examples of those spaces.

Multiplication operators on weighted Dirichlet spaces

Multiplication operators on weighted Dirichlet spaces

Carleson measures for Hardy-Sobolev spaces in the Siegel upper half-space

We give a capacitary type characterization of Carleson measures for a class of Hardy-Sobolev spaces (also known as weighted Dirichlet spaces) on the Siegel upper half-space, introduced by Arcozzi et al. in [7]. This answers in part a question raised by the same authors.

Generalized Cesàro operators on weighted Dirichlet spaces

Given a complex Borel measure η∈M(D), we study the boundedness of the Cesàro-type operator Cη given byCη(f)(z)=∑n=0∞(∫Dwndη(w))(∑k=0nak)zn where f(z)=∑n=0∞anzn, acting on the weighted Dirichlet spaces Dρ consisting in functions f∈H(D) with ∑n=0∞|an|2(n+1)ρn<∞ where (ρn) is a sequence of positive real numbers. We get new estimates on Taylor coefficients of functions in the Dirichlet space D0 and extend the results known for Dα and positive measures ν supported in [0,1) to more general weights and general measures.

Weighted composition operators that preserve p-frames

We establish the equivalence between being invertible and preserving p-frames for weighted composition operators on the unit disk. Moreover, we obtain the symbol properties of the bounded invertible operators. Based on those results, we prove that for the weighted Bergman spaces Aap(dAα), Besov spaces Bp and weighted Dirichlet spaces Dα2, weighted composition operators preserve p-frames on X if and only if they preserve q-Riesz bases on X⁎, and obtain related application on dynamical sampling of weighted composition operators. Furthermore, we characterize the equivalence between Fredholmness and the invertibility of weighted composition operators.

Inhomogeneous Poisson processes in the disk and interpolation

Abstract We investigate different geometrical properties, related to Carleson measures and pseudo-hyperbolic separation, of inhomogeneous Poisson point processes on the unit disk. In particular, we give conditions so that these random sequences are almost surely interpolating for the Hardy, Bloch or weighted Dirichlet spaces.

Measure induced Hankel and Toeplitz type operators on weighted Dirichlet spaces

For a complex Borel measure μ \mu on the open unit disk, and for a weighted Dirichlet space H s \mathcal {H}_s with 0 &gt; s &gt; 1 0&gt;s&gt;1 , we characterize the boundedness of the measure induced Hankel type operator H μ , s : H s → H s ¯ H_{\mu ,s}: \mathcal {H}_s \to \overline {\mathcal {H}_s} , extending the results of Xiao [Bull. Austral. Math. Soc. 62 (2000), pp. 135–140] for the classical Hardy space H 2 = H 1 H^2=\mathcal {H}_1 , and of Arcozzi, Rochberg, Sawyer, and Wick [J. Lond. Math. Soc. (2) 83 (2011), pp. 1–18] for the classical Dirichlet space D = H 0 \mathcal {D}= \mathcal {H}_0 . Our approach relies on some recent results about weak products of complete Nevanlinna-Pick reproducing kernel Hilbert spaces. We also include some related results on Hankel measures, Carleson measures, and Toeplitz type operators on weighted Dirichlet spaces H s \mathcal {H}_s , 0 &gt; s &gt; 1 0&gt;s&gt;1 .

Hausdorff operators on some classical spaces of analytic functions

Abstract In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators $$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$ on three important function spaces (i.e., derivative Hardy spaces, weighted Dirichlet spaces, and Bloch type spaces), which is a continuation of the previous works of Mirotin et al. Here, $\mu $ is a positive Radon measure, K is a $\mu $ -measurable function on the open unit disk $\mathbb {D}$ , and $\sigma _w(z)$ is the classical Möbius transform of $\mathbb {D}$ .

Embedding model and de Branges-Rovnyak spaces in Dirichlet spaces

In this paper we study embeddings between de Branges-Rovnyak spaces and harmonically weighted Dirichlet spaces in terms of the boundary spectrum of b and the support of the measure μ, by using elementary reproducing kernel estimates. We completely characterize the embedding between the model spaces and the local Dirichlet spaces , and we discuss some applications.

Spectral sets of generalized Hausdorff matrices on spaces of holomorphic functions on [formula omitted

Here, we study a family of bounded operators H, acting on Banach spaces of holomorphic functions X↪O(D), which are subordinated in terms of a C0-semigroup of weighted composition operators (vtCϕt), i.e., H=∫0∞vtCϕtdν(t) in the strong sense for some Borel measure ν. This family of operators extends the so-called generalized Hausdorff operators. We obtain the spectrum, point spectrum and essential spectrum of H under mild assumptions on (vtCϕt),ν and X. In particular, we obtain these spectral sets for a wide family of generalized Hausdorff operators acting on Hardy spaces, weighted Bergman spaces, weighted Dirichlet spaces and little Korenblum classes. The description for the spectra of the infinitesimal generator of (vtCϕt) is the key for our findings.

Universal composition operators on weighted Dirichlet spaces

Universal composition operators on weighted Dirichlet spaces

De Branges–Rovnyak spaces and local Dirichlet spaces of higher order

Abstract We discuss de Branges–Rovnyak spaces ℋ ⁢ ( b ) {\mathcal{H}(b)} generated by nonextreme and rational functions b and local Dirichlet spaces of order m introduced in [S. Luo, C. Gu and S. Richter, Higher order local Dirichlet integrals and de Branges–Rovnyak spaces, Adv. Math. 385 2021, Paper No. 107748]. In that paper, the authors characterized nonextreme b for which the operator Y = S | ℋ ⁢ ( b ) {Y=S|_{\mathcal{H}(b)}} , the restriction of the shift operator S on H 2 {H^{2}} to ℋ ⁢ ( b ) {\mathcal{H}(b)} , is a strict 2 ⁢ m {2m} -isometry and proved that such spaces ℋ ⁢ ( b ) {\mathcal{H}(b)} are equal to local Dirichlet spaces of order m. Here we give a characterization of local Dirichlet spaces of order m in terms of the m-th derivatives that is a generalization of a known result on local Dirichlet spaces. We also find explicit formulas for b in the case when ℋ ⁢ ( b ) {\mathcal{H}(b)} coincides with local Dirichlet space of order m with equality of norms. Finally, we prove a property of wandering vectors of Y analogous to the property of wandering vectors of the restriction of S to harmonically weighted Dirichlet spaces obtained in [D. Sarason, Harmonically weighted Dirichlet spaces associated with finitely atomic measures, Integral Equations Operator Theory 31 1998, 2, 186–213].

Frequent hypercyclicity of the tensor products of composition operators on weighted Dirichlet spaces

In this paper, we mainly study the frequent hypercyclicity of a scalar multiple of the tensor product of two linear fractional composition operators on , where is the weighted Dirichlet space. We will restrict the discussion to the situations that both φ and ψ are hyperbolic non automorphisms, hyperbolic automorphisms and parabolic automorphisms, respectively. Meanwhile, we give the equivalent conditions of to be frequently hypercyclic, hypercyclic, mixing and chaotic.

Asymptotics, Trace, and Density Results for Weighted Dirichlet Spaces Defined on the Half-line

We give an analytic description for the completion of C0∞(R+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_0^\\infty (\\mathbb {R}_+)$$\\end{document}, where R+=(0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}_+= (0,\\infty )$$\\end{document}, in Dirichlet space D1,p(R+,ω):={u:R+→R:u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{1,p}(\\mathbb {R}_+, \\omega ):= \\{ u \\, :\\mathbb {R}_+\\rightarrow {{\\mathbb {R}}}: u\\ $$\\end{document} is locally absolutely continuous on R+and‖u′‖Lp(R+,ω)<∞}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}_+\\, and \\, \\Vert u^{'}\\Vert _{L^p(\\mathbb {R}_+, \\omega )}<\\infty \\}$$\\end{document}, for given continuous positive weight ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} defined on R+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}_+$$\\end{document}, where 1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty $$\\end{document}. The conditions are described in terms of the modified variants of the Bp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_p$$\\end{document} conditions due to Kufner and Opic from 1984, which in our approach are focusing on the integrability of ω-p/(p-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega ^{-p/(p-1)}$$\\end{document} near zero or near infinity. Moreover, we propose applications of our results to: obtaining new variants of Hardy inequality, interpretation of boundary value problems in ODE’s defined on the half-line with solutions in D1,p(R+,ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{1,p}(\\mathbb {R}_+, \\omega )$$\\end{document}, new results from complex interpolation theory dealing with interpolation spaces between weighted Dirichlet spaces, and for deriving new Morrey type embedding theorems for our Dirichlet space.

Approximate spectral cosynthesis in the harmonically weighted Dirichlet spaces

For a finite positive Borel measure $\mu$ on the unit circle, let $\mathcal{D}(\mu)$ be the associated harmonically weighted Dirichlet space. A shift invariant subspace $\mathcal{M}$ recognizes strong approximate spectral cosynthesis if there exists a sequence of shift invariant subspaces $\mathcal{M}_k$, with finite codimension, such that the orthogonal projections onto $\mathcal{M}_k$ converge in the strong operator topology to the orthogonal projection onto $\mathcal{M}$. If $\mu$ is a finite sum of atoms, then we show that shift invariant subspaces of $\mathcal{D}(\mu)$ admit strong approximate spectral cosynthesis.

Volterra-type operators mapping weighted Dirichlet space into H^infty

The problem of describing the analytic functions g on the unit disc such that the integral operator T_g(f)(z)=int _0^zf(zeta )g'(zeta ),dzeta is bounded (or compact) from a Banach space (or complete metric space) X of analytic functions to the Hardy space H^infty is a tough problem, and remains unsettled in many cases. For analytic functions g with non-negative Maclaurin coefficients, we describe the boundedness and compactness of T_g acting from a weighted Dirichlet space D^p_omega , induced by an upper doubling weight omega , to H^infty . We also characterize, in terms of neat conditions on omega , the upper doubling weights for which T_g: D^p_omega rightarrow H^infty is bounded (or compact) only if g is constant.

Carleson’s formula for some weighted Dirichlet spaces

Abstract We extend Carleson’s formula to radially polynomially weighted Dirichlet spaces.

Composition operators on weighted analytic spaces

AbstractWe characterize the membership in the Schatten ideals $\mathcal {S}_p$ , $0&lt;p&lt;\infty $ , of composition operators acting on weighted Dirichlet spaces. Our results concern a large class of weights. In particular, we examine the case of perturbed superharmonic weights. Characterization of composition operators acting on weighted Bergman spaces to be in $\mathcal {S}_p$ is also given.

A new formalization of Dirichlet-type spaces

Applications and findings involving the norm of the weighted Dirichlet space Dα are steadily being discovered. There are notable characterizations of Dα through estimations of its norm. Applications include determining the conditions for boundedness of a Bergman type operator, and characterizing functions in the Drury–Arveson space that induce bounded Hankel operators. This paper improves the norm estimation by extending the domain of relevant parameters, and establishing the asymptotic formula with precise constants.

Weakly Multiplicative Distributions and Weighted Dirichlet Spaces

We show that if u is a compactly supported distribution on the complex plane such that, for every pair of entire functions f, g, $$\begin{aligned} \langle u,f{\overline{g}}\rangle =\langle u,f\rangle \langle u,{\overline{g}}\rangle , \end{aligned}$$ then u is supported at a single point. As an application, we complete the classification of all weighted Dirichlet spaces on the unit disk that are de Branges–Rovnyak spaces by showing that, for such spaces, the weight is necessarily a superharmonic function.