• Journal of the Korean Mathematical Society
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• v.40 no.3
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• pp.435-445
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• 2003

In this paper we consider Sobolev-type embedding theorems for harmonic and holomorphic Sobolev spaces on a bounded domain with $C^2$ boundary.

• Communications of the Korean Mathematical Society
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• v.14 no.1
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• pp.111-119
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• 1999

We consider the relations between locally convex spaces of generalized Sobolev spaces of Roumieu type.

• Journal of the Korean Mathematical Society
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• v.61 no.2
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• pp.341-356
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• 2024

In this paper, apply the regularities of the fractional Poisson kernels, we establish the Sobolev type trace inequalities of homogeneous Besov spaces, which are invariant under the conformal transforms. Also, by the aid of the Carleson measure characterizations of Q type spaces, the local version of Sobolev trace inequalities are obtained.

• Korean Journal of Mathematics
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• v.8 no.1
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• pp.73-86
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• 2000

We study the Sobolev spaces to the generalized Sobolev spaces $H^s_{\mathcal{G}}$ of exponential type based on the Silva space $\mathcal{G}$ and investigate its properties such as imbedding theorem and structure theorem. In fact, the imbedding theorem says that for $s$ > 0 $u{\in}H^s_{\mathcal{G}}$ can be analytically continued to the set {$z{\in}\mathbb{C}^n{\mid}{\mid}Im\;z{\mid}$ < $s$}. Also, the structure theorem means that for $s$ > 0 $u{\in}H^{-s}_{\mathcal{G}}$ is of the form $$u={\sum_{\alpha}\frac{s^{{|\alpha|}}}{{\alpha}!}D^{\alpha}g{\alpha}$$ where $g{\alpha}$'s are square integrable functions for ${\alpha}{\in}\mathbb{N}^n_0$. Moreover, we introduce a classes of symbols of exponential type and its associated pseudo-differential operators of exponential type, which naturally act on the generalized Sobolev spaces of exponential type. Finally, a generalized Bessel potential is defined and its properties are investigated.

• Korean Journal of Mathematics
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• v.16 no.1
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• pp.85-90
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• 2008

We consider Fourier multiplier operators whose symbols satisfy a generalization of $H{\ddot{o}}rmander^{\prime}s$ condition and establish their Sobolev-type mapping properties on the homogeneous Besov-Lipschitz spaces by making use of a continuous characterization of Besov-Lipschitz spaces. As an application, we derive Sobolev-type imbedding theorem.

• Journal of The Korean Astronomical Society
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• v.30 no.1
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• pp.13-25
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• 1997

Sobolev approximation can be adopted to a macroscopic supersonic motion comparatively larger than a random (thermal) one. It has recently been applied not only to the winds of hot early type stars, but also to envelopes of late type giants and/or supergiants. However, since the ratio of wind velocity to stochastic one is comparatively small in the winds of these stars, the condition for applying the Sobolev approximation is not fulfilled any more. Therefore the validity of the Sobolev approximation must be checked. We have calculated exact P Cygni profiles with various velocity ratios, $V_\infty/V_{sto}$, using the accelerated lambda iteration method, comparing with those obtained by the Sobolev approximation. While the velocity ratio decrease, serious deviations have been occured over the whole line profile. When the gradual increase in the velocity structure happens near the surface of star, the amount of deviations become more serious even at the high velocity ratios. The investigations have been applied to observed UV line profile of CIV in the Copernicus spectrums $of\;\zeta\;Puppis\;and\;NV\;of\;\tau\;Sco$. In case of $\tau$ Sco which has an expanding envelope with the gradual velocity increase in the inner region, The Sobolev approximation has given the serious deviations in the line profiles.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.603-617
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• 1997

Consider a Sobolev inner product on the space of polynomials such as $$ \phi(p,q) = \lambda p(c)q(c) + <\tau,p'(x)q'(x)> $$ where $\tau$ is a moment functional and c and $\lambda$ are real constants. We investigate properties of orthogonal polynomials relative to $\phi(\cdot,\cdot)$ and give necessary and sufficient conditions under which such Sobolev orthogonal polynomials satisfy a spectral type differential equation with polynomial coefficients.

• Bulletin of the Korean Mathematical Society
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• v.39 no.4
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• pp.561-569
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• 2002

In this paper we prove the existence and uniqueness of a mild solution of a functional differential equation of Sobolev type with nonlocal condition using the semigroup theory and the Banach fixed point principle.

• Journal of the Korean Mathematical Society
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• v.36 no.2
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• pp.257-266
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• 1999

We used Carleson and Chang's method to give another proof of the Moser-Trudinger inequality which was known as a limiting case of the Sobolev imbedding theorem and at the same time we get sharper information for the bound.

• Communications of the Korean Mathematical Society
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• v.14 no.1
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• pp.223-231
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• 1999

In this paper we prove the existence of solutions of a nonlinear integrodifferential equation of Sobolev type with nonlocal conditions. The results are obtained by using compact semigroups and the Schauder fixed point theorem.