• Title, Summary, Keyword: Dirichlet space

Search Result 52, Processing Time 0.045 seconds

Generalized Integration Operator between the Bloch-type Space and Weighted Dirichlet-type Spaces

  • Ardebili, Fariba Alighadr;Vaezi, Hamid;Hassanlou, Mostafa
    • Kyungpook Mathematical Journal
    • /
    • v.60 no.3
    • /
    • pp.519-534
    • /
    • 2020
  • Let H(𝔻) be the space of all holomorphic functions on the open unit disc 𝔻 in the complex plane ℂ. In this paper, we investigate the boundedness and compactness of the generalized integration operator $$I^{(n)}_{g,{\varphi}}(f)(z)=\normalsize\displaystyle\smashmargin{2}{\int\nolimits_0}^z\;f^{(n)}({\varphi}({\xi}))g({\xi})\;d{\xi},\;z{\in}{\mathbb{D}},$$ between Bloch-type and weighted Dirichlet-type spaces, where 𝜑 is a holomorphic self-map of 𝔻, n ∈ ℕ and g ∈ H(𝔻).

NOTES ON THE SPACE OF DIRICHLET TYPE AND WEIGHTED BESOV SPACE

  • Choi, Ki Seong
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.26 no.2
    • /
    • pp.393-402
    • /
    • 2013
  • For 0 < $p$ < ${\infty}$, ${\alpha}$ > -1 and 0 < $r$ < 1, we show that if $f$ is in the space of Dirichlet type $\mathfrak{D}^p_{p-1}$, then ${\int}_{1}^{0}M_{p}^{p}(r,f^{\prime})(1-r)^{p-1}rdr$ < ${\infty}$ and ${\int}_{1}^{0}M_{(2+{\alpha})p}^{(2+{\alpha})p}(r,f^{\prime})(1-r)^{(2+{\alpha})p+{\alpha}}rdr$ < ${\infty}$ where $M_p(r,f)=\[\frac{1}{2{\pi}}{\int}_{0}^{2{\pi}}{\mid}f(re^{it}){\mid}^pdt\]^{1/p}$. For 1 < $p$ < $q$ < ${\infty}$ and ${\alpha}+1$ < $p$, we show that if there exists some positive constant $c$ such that ${\parallel}f{\parallel}_{L^{q(d{\mu})}}{\leq}c{\parallel}f{\parallel}_{\mathfrak{D}^p_{\alpha}}$ for all $f{\in}\mathfrak{D}^p_{\alpha}$, then ${\parallel}f{\parallel}_{L^{q(d{\mu})}}{\leq}c{\parallel}f{\parallel}_{\mathcal{B}_p(q)}$ where $\mathcal{B}_p(q)$ is the weighted Besov space. We also find the condition of measure ${\mu}$ such that ${\sup}_{a{\in}D}{\int}_D(k_a(z)(1-{\mid}a{\mid}^2)^{(p-a-1)})^{q/p}d{\mu}(z)$ < ${\infty}$.

MULTIPLIERS OF DIRICHLET-TYPE SUBSPACES OF BLOCH SPACE

  • Li, Songxiao;Lou, Zengjian;Shen, Conghui
    • Bulletin of the Korean Mathematical Society
    • /
    • v.57 no.2
    • /
    • pp.429-441
    • /
    • 2020
  • Let M(X, Y) denote the space of multipliers from X to Y, where X and Y are analytic function spaces. As we known, for Dirichlet-type spaces 𝓓αp, M(𝓓p-1p, 𝓓q-1q) = {0}, if p ≠ q, 0 < p, q < ∞. If 0 < p, q < ∞, p ≠ q, 0 < s < 1 such that p + s, q + s > 1, then M(𝓓p-2+sp, 𝓓q-2+sq) = {0}. However, X ∩ 𝓓p-1p ⊆ X ∩ 𝓓q-1q and X ∩ 𝓓p-2+sp ⊆ X ∩ 𝓓q-2+sp whenever X is a subspace of the Bloch space 𝓑 and 0 < p ≤ q < ∞. This says that the set of multipliers M(X ∩ 𝓓 p-2+sp, X∩𝓓q-2+sq) is nontrivial. In this paper, we study the multipliers M(X ∩ 𝓓p-2+sp, X ∩ 𝓓q-2+sq) for distinct classical subspaces X of the Bloch space 𝓑, where X = 𝓑, BMOA or 𝓗.

DIRICHLET FORMS, DIRICHLET OPERATORS, AND LOG-SOBOLEV INEQUALITIES FOR GIBBS MEASURES OF CLASSICAL UNBOUNDED SPIN SYSTEM

  • Lim, Hye-Young;Park, Yong-Moon;Yoo, Hyun-Jae
    • Journal of the Korean Mathematical Society
    • /
    • v.34 no.3
    • /
    • pp.731-770
    • /
    • 1997
  • We study Diriclet forms and related subjects for the Gibbs measures of classical unbounded sping systems interacting via potentials which are superstable and regular. For any Gibbs measure $\mu$, we construct a Dirichlet form and the associated diffusion process on $L^2(\Omega, d\mu), where \Omega = (R^d)^Z^\nu$. Under appropriate conditions on the potential we show that the Dirichlet operator associated to a Gibbs measure $\mu$ is essentially self-adjoint on the space of smooth bounded cylinder functions. Under the condition of uniform log-concavity, the Gibbs measure exists uniquely and there exists a mass gap in the lower end of the spectrum of the Dirichlet operator. We also show that under the condition of uniform log-concavity, the unique Gibbs measure satisfies the log-Sobolev inequality. We utilize the general scheme of the previous works on the theory in infinite dimensional spaces developed by e.g., Albeverio, Antonjuk, Hoegh-Krohn, Kondratiev, Rockner, and Kusuoka, etc, and also use the equilibrium condition and the regularity of Gibbs measures extensively.

  • PDF

NONTRIVIAL PERIODIC SOLUTION FOR THE SUPERQUADRATIC PARABOLIC PROBLEM

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
    • /
    • v.17 no.1
    • /
    • pp.53-66
    • /
    • 2009
  • We show the existence of a nontrivial periodic solution for the superquadratic parabolic equation with Dirichlet boundary condition and periodic condition with a superquadratic nonlinear term at infinity which have continuous derivatives. We use the critical point theory on the real Hilbert space $L_2({\Omega}{\times}(0 2{\pi}))$. We also use the variational linking theorem which is a generalization of the mountain pass theorem.

  • PDF

A Semantic Aspect-Based Vector Space Model to Identify the Event Evolution Relationship within Topics

  • Xi, Yaoyi;Li, Bicheng;Liu, Yang
    • Journal of Computing Science and Engineering
    • /
    • v.9 no.2
    • /
    • pp.73-82
    • /
    • 2015
  • Understanding how the topic evolves is an important and challenging task. A topic usually consists of multiple related events, and the accurate identification of event evolution relationship plays an important role in topic evolution analysis. Existing research has used the traditional vector space model to represent the event, which cannot be used to accurately compute the semantic similarity between events. This has led to poor performance in identifying event evolution relationship. This paper suggests constructing a semantic aspect-based vector space model to represent the event: First, use hierarchical Dirichlet process to mine the semantic aspects. Then, construct a semantic aspect-based vector space model according to these aspects. Finally, represent each event as a point and measure the semantic relatedness between events in the space. According to our evaluation experiments, the performance of our proposed technique is promising and significantly outperforms the baseline methods.

GENERALIZED BOUNDED ANALYTIC FUNCTIONS IN THE SPACE Hω,p

  • Lee, Jun-Rak
    • Korean Journal of Mathematics
    • /
    • v.13 no.2
    • /
    • pp.193-202
    • /
    • 2005
  • We define a general space $H_{{\omega},p}$ of the Hardy space and improve that Aleman's results to the space $H_{{\omega},p}$. It follows that the multiplication operator on this space is cellular indecomposable and that each invariant subspace contains nontrivial bounded functions.

  • PDF

ON THE CLOSED RANGE COMPOSITION AND WEIGHTED COMPOSITION OPERATORS

  • Keshavarzi, Hamzeh;Khani-Robati, Bahram
    • Communications of the Korean Mathematical Society
    • /
    • v.35 no.1
    • /
    • pp.217-227
    • /
    • 2020
  • Let ψ be an analytic function on 𝔻, the unit disc in the complex plane, and φ be an analytic self-map of 𝔻. Let 𝓑 be a Banach space of functions analytic on 𝔻. The weighted composition operator Wφ,ψ on 𝓑 is defined as Wφ,ψf = ψf ◦ φ, and the composition operator Cφ defined by Cφf = f ◦ φ for f ∈ 𝓑. Consider α > -1 and 1 ≤ p < ∞. In this paper, we prove that if φ ∈ H(𝔻), then Cφ has closed range on any weighted Dirichlet space 𝒟α if and only if φ(𝔻) satisfies the reverse Carleson condition. Also, we investigate the closed rangeness of weighted composition operators on the weighted Bergman space Apα.

Nonparametric Bayesian Multiple Comparisons for Geometric Populations

  • Ali, M. Masoom;Cho, J.S.;Begum, Munni
    • Journal of the Korean Data and Information Science Society
    • /
    • v.16 no.4
    • /
    • pp.1129-1140
    • /
    • 2005
  • A nonparametric Bayesian method for calculating posterior probabilities of the multiple comparison problem on the parameters of several Geometric populations is presented. Bayesian multiple comparisons under two different prior/ likelihood combinations was studied by Gopalan and Berry(1998) using Dirichlet process priors. In this paper, we followed the same approach to calculate posterior probabilities for various hypotheses in a statistical experiment with a partition on the parameter space induced by equality and inequality relationships on the parameters of several geometric populations. This also leads to a simple method for obtaining pairwise comparisons of probability of successes. Gibbs sampling technique was used to evaluate the posterior probabilities of all possible hypotheses that are analytically intractable. A numerical example is given to illustrate the procedure.

  • PDF