• 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.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.481-494
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• 1996

We introduced the generalized Sobolev space $H_\omega^s$ in [4]. In this paper, we introduce the space $H_\omega^s(\Omega)$ of the generalized distributions in $H_\omega^s$ with compact supports in $\Omega$ and the local generalized Sobolev spaces $H_{\omega loc}^s(\Omega)$ of the generalized distributions on $\Omega$ which are locally in $H_\omega^s$ and study their properties.

• Korean Journal of Mathematics
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• v.29 no.1
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• pp.57-64
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• 2021

In this paper we deal with the embedding inclusions on the fractional Orlicz-Sobolev spaces which are crucial roles for studying the theories of the partial differential equations. We get some properties and theories of the embedding inclusions on the fractional Orlicz-Sobolev spaces.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.721-745
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• 2020

In this paper we study fractional Sobolev spaces characterized by a norm based on eigenfunction expansions. The goal of this paper is twofold. The first one is to define fractional Sobolev spaces of order -1 ≤ s ≤ 2 equipped with a norm defined in terms of Neumann eigenfunction expansions. Due to the zero Neumann trace of Neumann eigenfunctions on a boundary, fractional Sobolev spaces of order 3/2 ≤ s ≤ 2 characterized by the norm are the spaces of functions with zero Neumann trace on a boundary. The spaces equipped with the norm are useful for studying cross-sectional traces of solutions to the Helmholtz equation in waveguides with a homogeneous Neumann boundary condition. The second one is to define fractional Sobolev spaces of order -1 ≤ s ≤ 1 for vector-valued functions in a simply-connected, bounded and smooth domain in ℝ². These spaces are defined by a norm based on series expansions in terms of eigenfunctions of the vector Laplacian with boundary conditions of zero tangential component or zero normal component. The spaces defined by the norm are important for analyzing cross-sectional traces of time-harmonic electromagnetic fields in perfectly conducting waveguides.

• 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.

• 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.

• Journal of the Korean Mathematical Society
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• v.50 no.6
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• pp.1257-1269
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• 2013

Using the three critical points theorem by B. Ricceri [23], we obtain a multiplicity result for a class of nonlocal problems in Orlicz-Sobolev spaces. To our knowledge, this is the first contribution to the study of nonlocal problems in this class of functional spaces.

• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1143-1149
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• 2020

Sobolev trace inequalities on nonhomogeneous fractional Sobolev spaces are established.