• Title/Summary/Keyword: weighted Sobolev space

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On a weighted hardy-sobolev space functions (I)

  • Kwon, E.G.
    • Communications of the Korean Mathematical Society
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    • v.11 no.2
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    • pp.349-357
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    • 1996
  • Using a special property of Bloch functions with Hardmard gaps and using the geometric properties of the self maps of the unit disc, we give a way of constructing explicit examples of Bloch functions f whose derivative is in $H^p$ (0 < p < 1) but $f \notin BMOA$.

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New Two-Weight Imbedding Inequalities for $\mathcal{A}$-Harmonic Tensors

  • Gao, Hongya;Chen, Yanmin;Chu, Yuming
    • Kyungpook Mathematical Journal
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    • v.47 no.1
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    • pp.105-118
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    • 2007
  • In this paper, we first define a new kind of two-weight-$A_r^{{\lambda}_3}({\lambda}_1,{\lambda}_2,{\Omega})$-weight, and then prove the imbedding inequalities for $\mathcal{A}$-harmonic tensors. These results can be used to study the weighted norms of the homotopy operator T from the Banach space $L^p(D,{\bigwedge}^l)$ to the Sobolev space $W^{1,p}(D,{\bigwedge}^{l-1})$, $l=1,2,{\cdots},n$, and to establish the basic weighted $L^p$-estimates for $\mathcal{A}$-harmonic tensors.

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CENTRAL LIMIT TYPE THEOREM FOR WEIGHTED PARTICLE SYSTEMS

  • Cho, Nhan-Sook;Kwon, Young-Mee
    • Journal of the Korean Mathematical Society
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    • v.41 no.5
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    • pp.773-793
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    • 2004
  • We consider a system of particles with locations { $X_{i}$ $^{n}$ (t):t$\geq$0,i=1,…,n} in $R^{d}$ , time-varying weights { $A_{i}$ $^{n}$ (t) : t $\geq$0,i = 1,…,n} and weighted empirical measure processes $V^{n}$ (t)=1/n$\Sigma$$_{i=1}$$^{n}$ $A_{i}$ $^{n}$ (t)$\delta$ $X_{i}$ $^{n}$ (t), where $\delta$$_{x}$ is the Dirac measure. It is known that there exists the limit of { $V_{n}$ } in the week* topology on M( $R^{d}$ ) under suitable conditions. If { $X_{i}$ $^{n}$ , $A_{i}$ $^{n}$ , $V^{n}$ } satisfies some diffusion equations, applying Ito formula, we prove a central limit type theorem for the empirical process { $V^{n}$ }, i.e., we consider the convergence of the processes η$_{t}$ $^{n}$ ≡ n( $V^{n}$ -V). Besides, we study a characterization of its limit.t.

THE ${\bar{\partial}}$-PROBLEM WITH SUPPORT CONDITIONS AND PSEUDOCONVEXITY OF GENERAL ORDER IN KÄHLER MANIFOLDS

  • Saber, Sayed
    • Journal of the Korean Mathematical Society
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    • v.53 no.6
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    • pp.1211-1223
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    • 2016
  • Let M be an n-dimensional $K{\ddot{a}}hler$ manifold with positive holomorphic bisectional curvature and let ${\Omega}{\Subset}M$ be a pseudoconvex domain of order $n-q$, $1{\leq}q{\leq}n$, with $C^2$ smooth boundary. Then, we study the (weighted) $\bar{\partial}$-equation with support conditions in ${\Omega}$ and the closed range property of ${\bar{\partial}}$ on ${\Omega}$. Applications to the ${\bar{\partial}}$-closed extensions from the boundary are given. In particular, for q = 1, we prove that there exists a number ${\ell}_0$ > 0 such that the ${\bar{\partial}}$-Neumann problem and the Bergman projection are regular in the Sobolev space $W^{\ell}({\Omega})$ for ${\ell}$ < ${\ell}_0$.

STABILITY AND TOPOLOGY OF TRANSLATING SOLITONS FOR THE MEAN CURVATURE FLOW WITH THE SMALL Lm NORM OF THE SECOND FUNDAMENTAL FORM

  • Eungmo, Nam;Juncheol, Pyo
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.1
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    • pp.171-184
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    • 2023
  • In this paper, we show that a complete translating soliton Σm in ℝn for the mean curvature flow is stable with respect to weighted volume functional if Σ satisfies that the Lm norm of the second fundamental form is smaller than an explicit constant that depends only on the dimension of Σ and the Sobolev constant provided in Michael and Simon [12]. Under the same assumption, we also prove that under this upper bound, there is no non-trivial f-harmonic 1-form of L2f on Σ. With the additional assumption that Σ is contained in an upper half-space with respect to the translating direction then it has only one end.