• Title/Summary/Keyword: Dirichlet space

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DIRICHLET-JORDAN THEOREM ON $SIM$ SPACE

  • Kim, Hwa-Joon;Lekcharoen, S.;Supratid, S.
    • Korean Journal of Mathematics
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    • 제17권1호
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    • pp.37-41
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    • 2009
  • We would like to propose Dirichlet-Jordan theorem on the space of summable in measure(SIM). Surely, this is a kind of extension of bounded variation([1, 4]), and considered as an application of fuzzy set such that ${\alpha}$-cut is 0.

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THE CORONA THEOREM FOR BOUNDED FUNCTIONS IN DIRICHLET SPACE

  • Nah, Young-Chae
    • 충청수학회지
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    • 제10권1호
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    • pp.141-146
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    • 1997
  • In this paper we prove that the corona theorem for the algebra $H^{\infty}(D){\cap}D(D)$. That is, we prove that $\mathcal{M}{\setminus}{\overline{D}}$ is an empty set where $\mathcal{M}$ is the maximal ideal space of the given algebra.

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

  • Ardebili, Fariba Alighadr;Vaezi, Hamid;Hassanlou, Mostafa
    • Kyungpook Mathematical Journal
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    • 제60권3호
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    • pp.519-534
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    • 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).

AN ABSTRACT DIRICHLET PROBLEM IN THE HILBERT SPACE

  • Hamza-A.S.Abujabal;Mahmoud-M.El-Boral
    • Journal of applied mathematics & informatics
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    • 제4권1호
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    • pp.109-116
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    • 1997
  • In the present paper we consider an abstract partial dif-ferential equation of the form $\frac{\partial^2u}{{\partial}t^2}-\frac{\partial^2u}{{\partial}x^2}+A(x.t)u=f(x, t)$, where ${A(x, t):(x, t){\epsilon}\bar{G} }$ is a family of linear closed operators and $G=GU{\partial}G$, G is a suitable bounded region in the (x, t)-plane with bound-are ${\partial}G$. It is assumed that u is given on the boundary ${\partial}G$. The objective of this paper is to study the considered Dirichlet problem for a wide class of operators $A(x, t)$. A Dirichlet problem for non-elliptic partial differential equations of higher orders is also considerde.

NOTES ON THE SPACE OF DIRICHLET TYPE AND WEIGHTED BESOV SPACE

  • Choi, Ki Seong
    • 충청수학회지
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    • 제26권2호
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    • pp.393-402
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    • 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
    • 대한수학회보
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    • 제57권2호
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    • pp.429-441
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    • 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 EIGENVALUE PROBLEMS UNDER MUSIELAK-ORLICZ GROWTH

  • Benyaiche, Allami;Khlifi, Ismail
    • 대한수학회지
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    • 제59권6호
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    • pp.1139-1151
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    • 2022
  • This paper studies the eigenvalues of the G(·)-Laplacian Dirichlet problem $$\{-div\;\(\frac{g(x,\;{\mid}{\nabla}u{\mid})}{{\mid}{\nabla}u{\mid}}{\nabla}u\)={\lambda}\;\(\frac{g(x,{\mid}u{\mid})}{{\mid}u{\mid}}u\)\;in\;{\Omega}, \\u\;=\;0\;on\;{\partial}{\Omega},$$ where Ω is a bounded domain in ℝN and g is the density of a generalized Φ-function G(·). Using the Lusternik-Schnirelmann principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues.

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
    • 대한수학회지
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    • 제34권3호
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    • pp.731-770
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    • 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.

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NONTRIVIAL PERIODIC SOLUTION FOR THE SUPERQUADRATIC PARABOLIC PROBLEM

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • 제17권1호
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    • pp.53-66
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    • 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.

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GENERALIZED BOUNDED ANALYTIC FUNCTIONS IN THE SPACE Hω,p

  • Lee, Jun-Rak
    • Korean Journal of Mathematics
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    • 제13권2호
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    • pp.193-202
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    • 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.

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