• 대한수학회지
• /
• 제40권3호
• /
• pp.435-445
• /
• 2003

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

• 대한수학회논문집
• /
• 제14권1호
• /
• pp.111-119
• /
• 1999

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

• 대한수학회지
• /
• 제33권3호
• /
• pp.481-494
• /
• 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
• /
• 제29권1호
• /
• pp.57-64
• /
• 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.

• 대한수학회논문집
• /
• 제35권2호
• /
• pp.517-532
• /
• 2020

In this paper, we consider a class of nonlocal problems with indefinite weights in Orlicz-Sobolev space. Under some suitable conditions on the nonlinearities, we establish some existence results using variational techniques and Ekeland's variational principle.

• 대한수학회보
• /
• 제57권5호
• /
• pp.1143-1149
• /
• 2020

Sobolev trace inequalities on nonhomogeneous fractional Sobolev spaces are established.

• East Asian mathematical journal
• /
• 제15권1호
• /
• pp.19-28
• /
• 1999

• 대한수학회보
• /
• 제50권3호
• /
• pp.915-928
• /
• 2013

In this paper, we will provide an alternative proof to characterize the pointwise multipliers which maps a Sobolev space $\dot{H}^r(\mathb{R}^d)$ to its dual $\dot{H}^{-r}(\mathb{R}^d)$ in the case 0 < $r$ < $\frac{d}{2}$ by a simple application of the definition of fractional Sobolev space. The proof relies on a method introduced by Maz'ya-Verbitsky [9] to prove the same result.

• Korean Journal of Mathematics
• /
• 제8권1호
• /
• pp.73-86
• /
• 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 applied mathematics & informatics
• /
• 제23권1_2호
• /
• pp.435-443
• /
• 2007

An important approach towards solving the scattered data problem is by using radial basis functions. However, for a large class of smooth basis functions such as Gaussians, the existing theories guarantee the interpolant to approximate well only for a very small class of very smooth approximate which is the so-called 'native' space. The approximands f need to be extremely smooth. Hence, the purpose of this paper is to study approximation by a scattered shifts of a radial basis functions. We provide error estimates on larger spaces, especially on the homogeneous Sobolev spaces.