Dirichlet Space Research Articles

Reflections at infinity of time changed RBMs on a domain with Liouville branches

Let $Z$ be the transient reflecting Brownian motion on the closure of an unbounded domain $D\subset \mathbb{R}^d$ with $N$ number of Liouville branches. We consider a diffuion $X$ on $\overline{D}$ having finite lifetime obtained from $Z$ by a time change. We show that $X$ admits only a finite number of possible symmetric conservative diffusion extensions $Y$ beyond its lifetime characterized by possible partitions of the collection of $N$ ends and we identify the family of the extended Dirichlet spaces of all $Y$ (which are independent of time change used) as subspaces of the space $\mathrm{BL}(D)$ spanned by the extended Sobolev space $H_e^1(D)$ and the approaching probabilities of $Z$ to the ends of Liouville branches.

Difference of generalized integration operators on the space of Cauchy transforms

ABSTRACTIn 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.

On the Non‐Polyhedricity of Sets with Upper and Lower Bounds in Dual Spaces

AbstractWe demonstrate that the set L∞(X, [−1,1]) of all measurable functions over a Borel measure space (X, B, μ) with values in the unit interval is typically non‐polyhedric when interpreted as a subset of a dual space. Our findings contrast the classical result that subsets of Dirichlet spaces with pointwise upper and lower bounds are polyhedric. In particular, additional structural assumptions are unavoidable when the concept of polyhedricity is used to study the differentiability properties of solution maps to variational inequalities of the second kind in, e.g., the spaces H1/2(∂Ω) or H01(Ω).

Kernels and ranks of Hankel operators on the Dirichlet spaces

We consider Hankel operators and little Hankel operators on the Dirichlet space and pluriharmonic Dirichlet space of the unit ball. We first describe kernels of Hankel operators with pluriharmonic symbol. Next, we prove that there are no non-trivial finite rank Hankel operators with pluriharmonic symbol. Finally, we characterize finite rank little Hankel operators with pluriharmonic symbol. Our results show that there are several differences between the results on the Dirichlet space and pluriharmonic Dirichlet space.

Regularity of Gaussian Processes on Dirichlet Spaces

We study the regularity of centered Gaussian processes $$(Z_x( \omega ))_{x\in M}$$ , indexed by compact metric spaces $$(M, \rho )$$ . It is shown that the almost everywhere Besov regularity of such a process is (almost) equivalent to the Besov regularity of the covariance $$K(x,y) = {\mathbb E}(Z_x Z_y)$$ under the assumption that (i) there is an underlying Dirichlet structure on M that determines the Besov regularity, and (ii) the operator K with kernel K(x, y) and the underlying operator A of the Dirichlet structure commute. As an application of this result, we establish the Besov regularity of Gaussian processes indexed by compact homogeneous spaces and, in particular, by the sphere.

Representing systems generated by reproducing kernels

In this paper, we study the representing and absolutely representing systems in the context of reproducing kernel Hilbert spaces. We prove in particular that, in many classical spaces including weighted Hardy and Dirichlet spaces and de Branges–Rovnyak spaces, there cannot exist absolutely representing systems of reproducing kernels.

THE ESSENTIAL NORMS OF COMPOSITION OPERATORS ON WEIGHTED DIRICHLET SPACES

Let $\unicode[STIX]{x1D711}$ be an analytic self-map of the unit disc. If $\unicode[STIX]{x1D711}$ is analytic in a neighbourhood of the closed unit disc, we give a precise formula for the essential norm of the composition operator $C_{\unicode[STIX]{x1D711}}$ on the weighted Dirichlet spaces ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}$ for $\unicode[STIX]{x1D6FC}>0$. We also show that, for a univalent analytic self-map $\unicode[STIX]{x1D711}$ of $\mathbb{D}$, if $\unicode[STIX]{x1D711}$ has an angular derivative at some point of $\unicode[STIX]{x2202}\mathbb{D}$, then the essential norm of $C_{\unicode[STIX]{x1D711}}$ on the Dirichlet space is equal to one.

A boundary norm for weighted Dirichlet spaces

A boundary norm for weighted Dirichlet spaces

Reducing subspaces of non-analytic Toeplitz operators on weighted Hardy and Dirichlet spaces of the bidisk

We characterize reducing subspaces of a class of non-analytic Toeplitz operators on Hilbert spaces of holomorphic functions on the bidisk including the Hardy space and the Dirichlet space of the bidisk. These results compliment a recent characterization of the reducing subspaces for this class of non-analytic Toeplitz operators on the Bergman space of the bidisk.

A Gleason–Kahane–Żelazko theorem for the Dirichlet space

We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a characterization of weighted composition operators on the Dirichlet space as being exactly those linear maps that send nowhere-vanishing functions to nowhere-vanishing functions.We also investigate possible extensions to weighted Dirichlet spaces with superharmonic weights. As part of our investigation, we are led to determine which of these spaces contain functions that map the unit disk onto the whole complex plane.

Interpolating Sequences in Spaces with the Complete Pick Property

Abstract We characterize interpolating sequences for multiplier algebras of spaces with the complete Pick property. Specifically, we show that a sequence is interpolating if and only if it is separated and generates a Carleson measure. This generalizes results of Carleson for the Hardy space and of Bishop, Marshall, and Sundberg for the Dirichlet space. Furthermore, we investigate interpolating sequences for pairs of Hilbert function spaces.

Logarithmic and Linear Potentials of Signed Measures and Markov Property of Associated Gaussian Fields

We consider the family of finite signed measures on the complex plane \(\mathbb {C}\) with compact support, of finite logarithmic energy and with zero total mass. We show directly that the logarithmic potential of such a measure sits in the Beppo Levi space, namely, the extended Dirichlet space of the Sobolev space of order 1 over \(\mathbb {C}\), and that the half of its Dirichlet integral equals the logarithmic energy of the measure. We then derive the (local) Markov property of the Gaussian field \(\textbf {G}(\mathbb {C})\) indexed by this family of measures. Exactly analogous considerations will be made for the Beppo Levi space over the upper half plane \(\mathbb {H}\) and the Cameron-Martin space over the real line \(\mathbb {R}\). Some Gaussian fields appearing in recent literatures related to mathematical physics will be interpreted in terms of the present field \(\textbf {G}(\mathbb {C})\).

Invariant Subspaces for Classical Operators on Weighted Spaces of Holomorphic Functions

We develop an approach to describe invariant subspaces of the integration operator on various scales of weighted spaces of holomorphic functions on the unit disk and the complex plane. It allows us to solve the problem for wide classes of Bergman, Bloch, Dirichlet, and Fock spaces, while all previous known results concern spaces defined by some weights of a special form. In addition, we also show that an analogous method works as well for the differentiation operator on weighted spaces of entire functions.

Toeplitz Operator for Dirichlet Space Through Sobolev Multiplier Algebra

This paper is mainly concerned with the Toeplitz operator $T_{\phi}$ over the Dirichlet space $\mathcal{D}$ with the symbol $\phi$ in the Sobolev multiplier algebra $M(W^{1,2}(\mathbb{D}))$, thereby extending several known ones in a very different manner.

Dirichlet Spaces Associated with Locally Finite Rooted Directed Trees

Let $$\mathscr {T}=(V, \mathcal E)$$ be a leafless, locally finite rooted directed tree. We associate with $$\mathscr {T}$$ a one parameter family of Dirichlet spaces $$\mathscr {H}_q~(q \geqslant 1)$$ , which turn out to be Hilbert spaces of vector-valued holomorphic functions defined on the unit disc $$\mathbb D$$ in the complex plane. These spaces can be realized as reproducing kernel Hilbert spaces associated with the positive definite kernel $$\begin{aligned} \kappa _{\mathscr {H}_q}(z, w)= & {} \sum _{n=0}^{\infty }\frac{(1)_n}{(q)_n}\,{z^n \overline{w}^n} ~P_{\langle e_{\mathsf {root}}\rangle } \\&+\,\sum _{v \in V_{\prec }} \sum _{n=0}^{\infty } \frac{(n_v +2)_n}{(n_v + q+1)_n}\, {z^n \overline{w}^n}~P_{v}~(z, w \in \mathbb D), \end{aligned}$$ where $$V_{\prec }$$ denotes the set of branching vertices of $$\mathscr {T}$$ , $$n_v$$ denotes the depth of $$v \in V$$ in $$\mathscr {T},$$ and $$P_{\langle e_{\mathsf {root}}\rangle }$$ , $$~P_{v}~(v \in V_{\prec })$$ are certain orthogonal projections. Further, we discuss the question of unitary equivalence of operators $$\mathscr {M}^{(1)}_z$$ and $$\mathscr {M}^{(2)}_z$$ of multiplication by z on Dirichlet spaces $$\mathscr {H}_q$$ associated with directed trees $$\mathscr {T}_1$$ and $$\mathscr {T}_2$$ respectively.

Эквивалентность рекуррентности и лиувиллева свойства для симметричных форм Дирихле [На англ.

Given a symmetric Dirichlet form $(\mathcal{E},\mathcal{F})$ on a (non-trivial) $\sigma$-finite measure space $(E,\mathcal{B},m)$ with associated Markovian semigroup $\{T_{t}\}_{t\in(0,\infty)}$, we prove that $(\mathcal{E},\mathcal{F})$ is both irreducible and recurrent if and only if there is no non-constant $\mathcal{B}$-measurable function $u:E\to[0,\infty]$ that is \emph{$\mathcal{E}$-excessive}, i.e., such that $T_{t}u\leq u$ $m$-a.e.\ for any $t\in(0,\infty)$. We also prove that these conditions are equivalent to the equality $\{u\in\mathcal{F}_{e}\mid \mathcal{E}(u,u)=0\}=\mathbb{R}\mathbf{1}$, where $\mathcal{F}_{e}$ denotes the extended Dirichlet space associated with $(\mathcal{E},\mathcal{F})$. The proof is based on simple analytic arguments and requires no additional assumption on the state space or on the form. In the course of the proof we also present a characterization of the $\mathcal{E}$-excessiveness in terms of $\mathcal{F}_{e}$ and $\mathcal{E}$, which is valid for any symmetric positivity preserving form.

Derivatives of rational inner functions: geometry of singularities and integrability at the boundary

We analyze the singularities of rational inner functions (RIFs) on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative H p membership purely in terms of contact order, a measure of the rate at which the zero set of an RIF approaches the distinguished boundary of the bidisk. We also show that derivatives of RIFs with singularities fail to be in H p for p ⩾ 3 2 and that higher non-tangential regularity of an RIF paradoxically reduces the H p integrability of its derivative. We derive inclusion results for Dirichlet-type spaces from derivative inclusion for H p . Using Agler decompositions and local Dirichlet integrals, we further prove that a restricted class of RIFs fails to belong to the unweighted Dirichlet space.

Intersection of harmonically weighted Dirichlet spaces

In 1991, S. Richter introduced harmonically weighted Dirichlet spaces D(μ), motivated by his study of cyclic analytic two-isometries. In this paper, we consider ⋂μ∈PD(μ), the intersection of D(μ) spaces, where P is the family of Borel probability measures. Several function-theoretic characterizations of the Banach space ⋂μ∈PD(μ) are given. We also show that ⋂μ∈PD(μ) is located strictly between some classical analytic Lipschitz spaces and ⋂μ∈PD(μ) can be regarded as the endpoint case of analytic Morrey spaces in some sense.

Approximation of Invariant Subspaces in Some Dirichlet-Type Spaces

The Hilbert space \(\mathcal {D}_{2}\) is the space of all holomorphic functions f defined on the open unit disc \(\mathbb {D}\) such that \({f}^{'}\) is in the Hardy Hilbert space \(\mathbf {H}^2.\) In this paper, we prove that the invariant subspaces of \(\mathcal {D}_{2}\) with respect to multiplication operator \(M_{z}\) can be approximated with finite co-dimensional invariant subspaces. We also obtain a partial result in this direction for the classical Dirichlet space.

Weighted Bergman–Dirichlet and Bargmann–Dirichlet Spaces in High Dimension

In this paper, we consider and study the n-dimensional extension of the Bergman–Dirichlet and Bargmann–Dirichlet spaces introduced recently in El Hamyani et al. (Ann Glob Anal Geom 49(1):59–72, 2016). We give a complete description of the considered spaces, including the explicit closed formulas for their reproducing kernel functions. Moreover, we investigate their asymptotic behavior when the curvature goes to 0.