• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.429-441
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• 2020

Let M(X, Y) denote the space of multipliers from X to Y, where X and Y are analytic function spaces. As we known, for Dirichlet-type spaces 𝓓_α^p, M(𝓓_p-1^p, 𝓓_q-1^q) = {0}, if p ≠ q, 0 < p, q < ∞. If 0 < p, q < ∞, p ≠ q, 0 < s < 1 such that p + s, q + s > 1, then M(𝓓_p-2+s^p, 𝓓_q-2+s^q) = {0}. However, X ∩ 𝓓_p-1^p ⊆ X ∩ 𝓓_q-1^q and X ∩ 𝓓_p-2+s^p ⊆ X ∩ 𝓓_q-2+s^p whenever X is a subspace of the Bloch space 𝓑 and 0 < p ≤ q < ∞. This says that the set of multipliers M(X ∩ 𝓓 _p-2+s^p, X∩𝓓_q-2+s^q) is nontrivial. In this paper, we study the multipliers M(X ∩ 𝓓_p-2+s^p, X ∩ 𝓓_q-2+s^q) for distinct classical subspaces X of the Bloch space 𝓑, where X = 𝓑, BMOA or 𝓗^∞.

• Journal of the Korean Geophysical Society
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• v.3 no.3
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• pp.193-200
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• 2000

The attenuation characteristics($Q^{-1}$) are important factors representing the physical properties of the Earth interiors, and are essential for the quantitative prediction of strong ground-motion. Based on 156 earthquakes including 76 single-station record on the seismic station located Deok-jung Ri, southeastern Korea, we made the simultaneous measurement of P and S wave attenuation($Q_P^{-1}\;and\;Q_S^{-1}$) by means of extended coda-normalization method. Estimated $Q_P^{-1}\;and\;Q_S^{-1}$ decreased from $1{\times}10^{-2}\;and\;9{\times}10^{-3}$ at 1.5 Hz to $6{\times}10^{-4}\;and\;5{\times}10^{-4}$ at 24 Hz, respectively. This can be expressed by $Q_P^{-1}=0.01\;f^{-1.07}\;and\;Q_S^{-1}=0.01\;f^{-1.03}$ which indicate strong frequency dependence.

• East Asian mathematical journal
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• v.19 no.1
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• pp.133-150
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• 2003

This paper will show that the relation (1.1) $$L^1({\Omega}){\subset}C_0(\bar{\Omega}){\subset}H_{p,q}$$ if 1/p'-1/n(1-2/q')<0 where p'=p/(p-1) and q'=q/(q-1) where $H_{p.q}=(W^{1,p}_0,W^{-1,p})_{1/q,q}$. We also intend to investigate the control problems for the retarded systems with $L^1(\Omega)$-valued controller in $H_{p,q}$.

• Journal of the Korean earth science society
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• v.27 no.4
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• pp.475-485
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• 2006

The physical properties of the central and southwestern crust of South Korea were estimated by comparing values of ${Q_P}^{-1}\;and\;{Q_S}^{-1}$ in the Kimcheon and Mokpo areas. In order to get ${Q_P}^{-1}\;and\;{Q_S}^{-1}$ values, seismic data were collected from two stations of the KIGAM network (KMC and MUN) and four stations of the KMA network (CPN, KUC, MOP, and WAN). An extended coda-normalization method was applied to these data. Estimates of ${Q_P}^{-1}\;and\;{Q_S}^{-1}$ show variations depending on frequency. As frequencies vary from 3 Hz to 24 Hz, the estimates decrease from $(1.4{\pm}3.9){\times}10^{-3}\;to\;(2.3{\pm}3.5){\times}10^{-4}\;for\;{Q_P}^{-1}\;and\;(1.8{\pm}1.3){\times}10^{-3}\;to\;(1.9{\pm}1.5){\times}10^{-4}\;for\;{Q_S}^{-1}$ in central South Korea, and $(5.9{\pm}4.8){\times}10^{-3}\;to\;(2.2{\pm}3.8){\times}10^{-4}\;for\;{Q_P}^{-1}\;and\;(0.5{\pm}2.8){\times}10^{-3}\;to\;(1.8{\pm}1.6){\times}10^{-4}\;for\;{Q_S}^{-1}$ in southwestern South Korea. According that a frequency-dependent power law is applied to the data, the best fits of ${Q_P}^{-1}\;and\;{Q_S}^{-1}\;are\;0.003f^{-0.49}\;and\;0.005f^{-1.03}$ in central South Korea, and $0.026f^{-1.47}$ and $0.001f^{-0.49}$ in southwestern South Korea, respectively. These values almost correspond to those of seismically stable regions although ${Q_P}^{-1}$ values of southwestern South Korea are a little high due to lack of data used.

• Journal of the Korean earth science society
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• v.22 no.6
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• pp.500-511
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• 2001

For the southeastern Korea aound the Yangsan fault we measured Q$_P^{-1}$ and Q$_S^{-1}$ simultaneously by using the extended coda-normalization method for seismograms registered at 9 stations deployed by KIGAM. We analyzed 707 seismograms of local earthquakes that occurred between December 1994 and February 2000. From seismograms, bandpass filtered traces were made by applying Butterworth filter with frequency-bands of 1${\sim}$2, 2${\sim}$4, 4${\sim}$8, 8${\sim}$16 and 16${\sim}$32 Hz. Estimated Q$_P^{-1}$ and Q$_S^{-1}$ values decrease from (7${\pm}$2)${\times}$10$^{-3}$ and (5${\pm}$4)${\times}$10$^{-4}$ at 1.5 Hz to (5${\pm}$4)${\times}$10$^{-3}$ and (5${\pm}$2)${\times}$10$^{-4}$ at 24 Hz, respectively. By fitting a power-law frequency dependent to estimated values over the whole stations, we obtained 0.009 (${\pm}$0.003)f$^{-1.05({\pm}0.14)$ for Q$_P^{-1}$ and 0.004 (${\pm}$0.001)f$^{-0.75({\pm}0.14)$) for Q$_S^{-1}$, where f is frequency in Hz.

• Journal of the Korean earth science society
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• v.28 no.6
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• pp.720-732
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• 2007

For the Seoul metropolitan area and the eastern Kyeongsang Basin, we simultaneously calculated $Q_P^{-1}$ and $Q_S^{-1}$ by applying the extended coda-normalization method for 98 seismograms of local Earthquakes. As frequency increases from 1.5 Hz to 24 Hz, the result decreased from $(4.0{\pm}9.2){\times}10^{-3}$ to $(4.1{\pm}4.2){\times}10^{-4}$ for $Q_P^{-1}$ and $(5.5{\pm}5.6){\times}10^{-3}$ to $(3.4{\pm}1.3){\times}10^{-4}$ for $Q_S^{-1}$ in Seoul Metropolitan Area. The result of eastern Kyeongsang basin also decreased from $(5.4{\pm}8.8){\times}10^{-3}$ to $(3.7{\pm}3.4){\times}10^{-4}$ for $Q_P^{-1}$ and $(5.7{\pm}4.2){\times}10^{-3}$ to $(3.5{\pm}1.6){\times}10^{-4}$ for $Q_S^{-1}$. If we fit a frequency-dependent power law to the data, the best fits of $Q_P^{-1}$ and $Q_S^{-1}$ are $0.005f^{-0.89}$ and $0.004f^{-0.88}$ in Seoul metropolitan Area, respectively. The value of $Q_P^{-1}$ and $Q_S^{-1}$ in the eastern Kyeongsang basin are $0.007f^{-1.02}$ and $0.006f^{-0.99}$, respectively. The $Q_S^{-1}$ value of the eastern Kyeongsang basin is almost similar to Seoul metropolitan area. But the $Q_P^{-1}$ value of the eastern Kyeongsang basin is a little higher than that of Seoul metropolitan area. This may be that the crustal characteristics of the eastern Kyeongsang basin is seismologically more heterogeneous. However, these $Q_P^{-1}$ values in Korea belong to the range of seismically stable regions all over the world.

• Journal of the Korean earth science society
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• v.22 no.2
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• pp.112-119
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• 2001

The Yangsan fault in the southeastern Korea has been receiving increasing attention in its seismic activity. In this fault region, by using the extended coda-normalization method for 707 seismograms of local earthquakes, were obtained 0.009f$^{-1.05}$ and 0.004f$^{-0.70}$ for fitting values of Q$_p^{-1}$ and Q$_s^{-1}$, respectively. These results indicate that Q$_p^{-1}$ and Q$_s^{-1}$ in the southeastern Korea is the lowest level in the world although the exponent values agree well with those in the other areas. The low Q-1 is not related to the movement of the Yangsan fault but to the tectonically inactive status like a shield area.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1181-1188
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• 2010

q-Volkenborn integrals ([8]) and fermionic invariant q-integrals ([12]) are introduced by T. Kim. By using these integrals, Euler q-zeta functions are introduced by T. Kim ([18]). Then, by using the Euler q-zeta functions, S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin ([25]) studied q-Genocchi zeta functions. And also Y. H. Kim, W. Kim, and C. S. Ryoo ([7]) investigated twisted q-zeta functions and their applications. In this paper, we consider the q-analogue of twisted Lerch type Euler zeta functions defined by $${\varsigma}E,q,\varepsilon(s)=[2]q \sum\limits_{n=0}^\infty\frac{(-1)^n\epsilon^nq^{sn}}{[n]_q}$$ where 0 < q < 1, $\mathfrak{R}$(s) > 1, $\varepsilon{\in}T_p$, which are compared with Euler q-zeta functions in the reference ([18]). Furthermore, we give the q-extensions of the above twisted Lerch type Euler zeta functions at negative integers which interpolate twisted q-Euler polynomials.

• Nonlinear Functional Analysis and Applications
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• v.26 no.2
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• pp.331-345
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• 2021

Let p(z)be a polynomial of degree n. Then Bernstein's inequality [12,18] is $${\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p^{\prime}(z){\mid}\;{\leq}\;n\;{\max_{{\mid}z{\mid}=1}{\mid}(z){\mid}}$$. For q > 0, we denote $${\parallel}p{\parallel}_q=\{{\frac{1}{2{\pi}}}{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{0}}^{2{\pi}}}\;{\mid}p(e^{i{\theta}}){\mid}^qd{\theta}\}^{\frac{1}{q}}$$, and a well-known fact from analysis [17] gives $${{\lim_{q{\rightarrow}{{\infty}}}}\{{\frac{1}{2{\pi}}}{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{0}}^{2{\pi}}}\;{\mid}p(e^{i{\theta}}){\mid}^qd{\theta}\}^{\frac{1}{q}}={\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p(z){\mid}$$. Above Bernstein's inequality was extended by Zygmund [19] into L^q norm by proving ║p'║_q ≤ n║p║_q, q ≥ 1. Let p(z) = a₀ + ∑ⁿ_𝜈=𝜇 a_𝜈z^𝜈, 1 ≤ 𝜇 ≤ n, be a polynomial of degree n having no zero in |z| < k, k ≥ 1. Then for 0 < r ≤ R ≤ k, Aziz and Zargar [4] proved $${\max\limits_{{\mid}z{\mid}=R}}\;{\mid}p^{\prime}(z){\mid}\;{\leq}\;{\frac{nR^{{\mu}-1}(R^{\mu}+k^{\mu})^{{\frac{n}{\mu}}-1}}{(r^{\mu}+k^{\mu})^{\frac{n}{\mu}}}\;{\max\limits_{{\mid}z{\mid}=r}}\;{\mid}p(z){\mid}}$$. In this paper, we obtain the L^q version of the above inequality for q > 0. Further, we extend a result of Aziz and Shah [3] into L^q analogue for q > 0. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.

• Kyungpook Mathematical Journal
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• v.63 no.2
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• pp.235-250
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• 2023

Let 1 ≤ p, q < ∞ be fixed, and let R = [r_jk] be an infinite scalar matrix such that 1 ≤ r_jk < ∞ and sup_j,k r_jk < ∞. Let 𝓑(𝑙_p, 𝑙_q) be the set of all bounded linear operator from 𝑙_p into 𝑙_q. For a fixed Banach algebra 𝐁 with identity, we define a new vector space S^R_p,q(𝐁) of infinite matrices over 𝐁 and a paranorm G on S^R_p,q(𝐁) as follows: let $$S^R_{p,q}({\mathbf{B}})=\{A:A^{[R]}{\in}{\mathcal{B}}(l_p,l_q)\}$$ and $G(A)={\parallel}A^{[R]}{\parallel}^{\frac{1}{M}}_{p,q}$, where $A^{[R]}=[{\parallel}a_{jk}{\parallel}^{r_{jk}}]$ and M = max{1, sup_j,k r_jk}. The existance of S^R_p,q(𝐁) equipped with the paranorm G(·) including its completeness are studied. We also provide characterizations of β -dual of the paranormed space.