• Communications of the Korean Mathematical Society
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• v.11 no.4
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• pp.1015-1030
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• 1996

We investigate complemented Banach subspaces of the Banach envelope of $eak L_1$. In particular, the Banach envelope of $weak L_1$ contains complemented Banach sublattices that are isometrically isomorphic to $l_p, (1 \leq p < \infty)$ or $c_0$. Finally, we also prove that the Banach envelope of $weak L_1$ contains an isomorphic copy of $l^{p, \infty}, (1 < p < \infty)$.

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.465-479
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• 2012

For $0<p{\leq}{\infty}$ and a convex body $K$ in $\mathbb{R}^n$, Lutwak, Yang and Zhang defined the concept of dual $L_p$-centroid body ${\Gamma}_{-p}K$ and $L_p$-John ellipsoid $E_pK$. In this paper, we prove the following two results: (i) For any origin-symmetric convex body $K$, there exist an ellipsoid $E$ and a parallelotope $P$ such that for $1{\leq}p{\leq}2$ and $0<q{\leq}{\infty}$, $E_qE{\supseteq}{\Gamma}_{-p}K{\supseteq}(nc_{n-2,p})^{-\frac{1}{p}}E_qP$ and $V(E)=V(K)=V(P)$; For $2{\leq}p{\leq}{\infty}$ and $0<q{\leq}{\infty}$, $2^{-1}{\omega_n}^{\frac{1}{n}}E_qE{\subseteq}{\Gamma}_{-p}K{\subseteq}{2\omega_n}^{-\frac{1}{n}}(nc_{n-2,p})^{-\frac{1}{p}}E_qP$ and $V(E)=V(K)=V(P)$. (ii) For any convex body $K$ whose John point is at the origin, there exists a simplex $T$ such that for $1{\leq}p{\leq}{\infty}$ and $0<q{\leq}{\infty}$, ${\alpha}n(nc_{n-2,p})^{-\frac{1}{p}}E_qT{\supseteq}{\Gamma}_{-p}K{\supseteq}(nc_{n-2,p})^{-\frac{1}{p}}E_qT$ and $V(K)=V(T)$.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.2
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• pp.269-278
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• 2008

Let ${\partial}D$ be the boundary of the open unit disk D in the complex plane and $L^p({\partial}D)$ the class of all complex, Lebesgue measurable function f for which $\{\frac{1}{2\pi}{\int}_{-\pi}^{\pi}{\mid}f(\theta){\mid}^pd\theta\}^{1/p}<{\infty}$. Let P be the orthogonal projection from $L^p({\partial}D)$ onto ${\cap}_{n<0}$ ker $a_n$. For $f{\in}L^1({\partial}D)$, ${\hat{f}}(z)=\frac{1}{2\pi}{\int}_{-\pi}^{\pi}P_r(t-\theta)f(\theta)d{\theta}$ is the harmonic extension of f. Let ${\hat{P}}$ be the composition of P with the harmonic extension. In this paper, we will show that if $1, then ${\hat{P}}:L^p({\partial}D){\rightarrow}H^p(D)$ is bounded. In particular, we will show that ${\hat{P}}$ is unbounded on $L^{\infty}({\partial}D)$.

• Kyungpook Mathematical Journal
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• v.53 no.1
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• pp.117-124
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• 2013

We show that for 1 < $p$ < ${\infty}$, $k$, $m{\in}\mathbb{N}$, $n^{(k)}(l_p)=inf\{n^{(k)}(l^m_p):m{\in}\mathbb{N}\}$ and that for any positive measure ${\mu}$, $n^{(k)}(L_p({\mu})){\geq}n^{(k)}(l_p)$. We also prove that for every $Q{\in}P(^kl_p:l_p)$ (1 < $p$ < ${\infty}$), if $v(Q)=0$, then ${\parallel}Q{\parallel}=0$.

• Journal of the Korean Mathematical Society
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• v.39 no.1
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• pp.61-75
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• 2002

Under the cancellation property and the Lipschitz condition on kernels, we prove that the Marcinkiewicz integrals defined on a homogeneous group H are bounded from $H^1$(H) to $L^1$(H), from $L_{c}$ $^{\infty}$(H) to BMO (H), and from $L^{p}$ (H) to $L^{p}$ (H) for 1 < p < $\infty$ assuming the $L^{q}$ -boundedness for some q > 1.for some q > 1.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.4 no.1
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• pp.67-77
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• 2000

In this paper we consider finite element mathods for nonlinear parabolic problems defined in ${\Omega}{\subset}R^d$ ($d{\leq}4$). A new initial approximation is taken. Optimal order error estimates in $L_p$ for $2{\leq}p{\leq}{\infty}$ are established for arbitrary order finite element. One order superconvergence in $W^{1,p}$ for $2{\leq}q{\leq}{\infty}$ are demonstrated as well.

• Journal of applied mathematics & informatics
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• v.38 no.5_6
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• pp.591-600
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• 2020

This work is a continuation of our investigations for p-adic analogue of the alternating form Dirichlet L-functions $$L_E(s,{\chi})={\sum\limits_{n=1}^{\infty}}{\frac{(-1)^n{\chi}(n)}{n^s}},\;Re(s)>0$$. Let L_p,E(s, t; χ) be the p-adic Euler L-function of two variables. In this paper, for any α ∈ ℂ_p, |α|_p ≤ 1, we give a power series expansion of L_p,E(s, t; χ) in terms of the variable t. From this, we derive a power series expansion of the generalized Euler polynomials with negative index, that is, we prove that $$E_{-n,{\chi}}(t)={\sum\limits_{m=0}^{\infty}}\(\array{-n\\m}\)E_{-(m+n),{\chi}^{t^m}},\;n{\in}{\mathbb{N}}$$, where t ∈ ℂ_p with |t|_p < 1. Some further properties for L_p,E(s, t; χ) has also been shown.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.2
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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 ${\sup}_{a{\in}D}{\int}_D(k_a(z)(1-{\mid}a{\mid}^2)^{(p-a-1)})^{q/p}d{\mu}(z)$ < ${\infty}$.

• Bulletin of the Korean Mathematical Society
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• v.36 no.3
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• pp.589-597
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• 1999

Let G be a closed subspace of a Banach space X and let (S,$\Omega$,$\mu$) be a $\sigma$-finite measure space. It was known that $L_1$(S,G) is proximinal in $L_1$(S,X) if and only if $L_p$(S,G) is proximinal in $L_p$(S,X) for 1$\infty$. In this article we show that this result remains true when "proximinal" is replaced by "Chebyshev". In addition, it is shown that if G is a proximinal subspace of X such that either G or the kernel of the metric projection $P_G$ is separable then, for 0 < p $\leq$ $\infty$. $L_p$(S,G) is proximinal in $L_p$(S,X)

• Communications of the Korean Mathematical Society
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• v.19 no.4
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• pp.715-720
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• 2004

Suppose X is a closed subspace of Z = ${({{\Sigma}^{\infty}}_{n=1}Z_{n})}_{p}$ (1 < p < ${\infty}$, dim $Z_{n}$ < ${\infty}$). We investigate an isometrically isomorphic embedding of L(X)/K(X) into L(X, Z)/K(X, Z), where L(X, Z) (resp. L(X)) is the space of the bounded linear operators from X to Z (resp. from X to X) and K(X, Z) (resp. K(X)) is the space of the compact linear operators from X to Z (resp. from X to X).