Abstract : In this paper, we are devoted to studying the mixed radial-angular integrabilities for Hardy type operators. As an application, the upper and lower bounds are obtained for the fractional Hardy operator. In addition, we also establish the sharp weak-type estimate for the fractional Hardy operator.
Abstract : In this paper, we establish the boundedness and continuity for variation operators for $\theta$-type Calder\'{o}n--Zygmund singular integrals and their commutators on the Triebel--Lizorkin spaces. As applications, we obtain the corresponding results for the Hilbert transform, the Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators.
Abstract : In this paper, we give an improvement for the second main theorems of algebraically non-degenerate meromorphic maps from generalized $ p $-parabolic manifolds into projective varieties intersecting hypersurfaces in subgeneral position with some index, which extends the results of Han [6] and Chen-Thin [3].
Abstract : In this paper, we prove some vanishing theorems under the assumptions of weighted BiRic curvature or $m$-Bakry-\'{E}mery-Ricci curvature bounded from below.
Abstract : We explicitly construct the smooth toric Fano variety which is isomorphic to the blow-up of the projective space at torus invariant points in codimension one by anti-flips.
Abstract : Let $F$ be a periodic singular fiber of genus $g$ with dual fiber $F^*$, and let $T$ (resp.~$T^*$) be the set of the components of $F$ (resp.~$F^*$) by removing one component with multiplicity one. We give a formula to compute the determinant $|\det T\,|$ of the intersect form of $T$. As a consequence, we prove that $|\det T\,|=|\det T^*\,|$. As an application, we compute the Mordell-Weil group of a fibration $f:S\to \mathbb P^1$ of genus $2$ with two singular fibers.
Abstract : In this study, we deal with American lookback option prices on dividend-paying assets under a stochastic volatility (SV) model. By using the asymptotic analysis introduced by Fouque et al. [17] and the Laplace-Carson transform (LCT), we derive the explicit formula for the option prices and the free boundary values with a finite expiration whose volatility is driven by a fast mean-reverting Ornstein-Uhlenbeck process. In addition, we examine the numerical implications of the SV on the American lookback option with respect to the model parameters and verify that the obtained explicit analytical option price has been obtained accurately and efficiently in comparison with the price obtained from the Monte-Carlo simulation.
Abstract : In 2010, Li-Ye [13, Theorem 0.1] proved that \begin{equation}\nonumber P\left(\zeta(z),\zeta'(z),\ldots,\zeta^{(m)}(z),\Gamma(z),\Gamma'(z),\Gamma^{''}(z)\right) \not\equiv 0\quad\text{in }\ \mathbb{C}, \end{equation} where $m$ is a non-negative integer, and $P(u_{0},u_{1}, \ldots, u_{m},v_{0},v_{1},v_{2})$ is any non-trivial polynomial in its arguments with coefficients in the field $\mathbb{C}$. Later on, Li-Ye [15, Theorem 1] proved that \begin{equation}\nonumber P\left(z,\Gamma(z),\Gamma'(z),\ldots,\Gamma^{(n)}(z), \zeta(z)\right)\not\equiv 0 \end{equation} in $z\in \mathbb{C}$ for any non-trivial distinguished polynomial $P(z,u_0, u_1,\ldots$, $u_n, v)$ with coefficients in a set $L_\delta$ of the zero function and a class of non-zero functions $f$ from $\mathbb{C}$ to $\mathbb{C}\cup\{\infty\}$ (cf. [15, Definition 1]). In this paper, we prove that $P\left(z,\zeta(z),\zeta'(z),\ldots,\zeta^{(m)}(z),\Gamma(z),\Gamma'(z),\ldots,\Gamma^{(n)}(z)\right)\not\equiv 0$ in $z\in\mathbb{C}$, where $m$ and $n$ are two non-negative integers, and $$P(z, u_0,u_1,\ldots,u_m,v_0,v_1,\ldots,v_n)$$ is any non-trivial polynomial in the $m+n+2$ variables $$u_0,u_1,\ldots,u_m,v_0,v_1,\ldots,v_n$$ with coefficients being meromorphic functions of order less than one, and the polynomial $P(z, u_0,u_1,\ldots,u_m,v_0,v_1,\ldots,v_n)$ is a distinguished polynomial in the $n+1$ variables $v_0,v_1,\ldots, v_n$. The question studied in this paper is concerning the conjecture of Markus from [16]. The main results obtained in this paper also extend the corresponding results from Li-Ye [12] and improve the corresponding results from Chen-Wang [5] and Wang-Li-Liu-Li [23], respectively.
Abstract : Inspired by Hongjie Dong and Qi S. Zhang's article [3], we find that the analyticity in time for a smooth solution of the heat equation with exponential quadratic growth in the space variable can be extended to any complete noncompact Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded below and the potential function being of at most quadratic growth. Therefore, our result holds on all gradient Ricci solitons. As a corollary, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with the similar growth condition. In addition, we also consider the solution in certain $L^p$ spaces with $p\in[2,+\infty)$ and prove its analyticity with respect to time.
Abstract : A Hilbert space operator $A\in{\mathcal B(H)}$ is a generalised \linebreak $n$-projection, denoted $A\in (G-n-P)$, if ${A^*}^n=A$. $(G-n-P)$-operators $A$ are normal operators with finitely countable spectra $\sigma(A)$, subsets of the set $\{0\}\cup\{\sqrt[n+1]{1}\}$. The Aluthge transform $\tilde{A}$ of $A\in{\mathcal B(H)}$ may be $(G-n-P)$ without $A$ being $(G-n-P)$. For doubly commuting operators $A, B\in{\mathcal B(H)}$ such that $\sigma(AB)=\sigma(A)\sigma(B)$ and $\|A\|\|B\|\leq \left\|\widetilde{AB}\right\|$, $\widetilde{AB}\in (G-n-P)$ if and only if $A=\left\|\tilde{A}\right\|(A_{00}\oplus(A_{0}\oplus A_u))$ and $B=\left\|\tilde{B}\right\|(B_0\oplus B_u)$, where $A_{00}$ and $B_0$, and $A_0\oplus A_u$ and $B_u$, doubly commute, $A_{00}B_0$ and $A_0$ are 2 nilpotent, $A_u$ and $B_u$ are unitaries, $A^{*n}_u=A_u$ and $B^{*n}_u=B_u$. Furthermore, a necessary and sufficient condition for the operators $\alpha A$, $\beta B$, $\alpha \tilde{A}$ and $\beta \tilde{B}$, $\alpha=\frac{1}{\left\|\tilde{A}\right\|}$ and $\beta=\frac{1}{\left\|\tilde{B}\right\|}$, to be $(G-n-P)$ is that $A$ and $B$ are spectrally normaloid at $0$.
Enkhbayar Azjargal, Zorigt Choinkhor, Nyamdavaa Tsegmid
Bull. Korean Math. Soc. 2023; 60(4): 1131-1139
https://doi.org/10.4134/BKMS.b220595
Chang Heon Kim, Namhun Koo , Soonhak Kwon
Bull. Korean Math. Soc. 2022; 59(6): 1523-1537
https://doi.org/10.4134/BKMS.b210870
Jun Ho Lee
Bull. Korean Math. Soc. 2023; 60(2): 315-323
https://doi.org/10.4134/BKMS.b220094
Kanchan Jangra, Dinesh Udar
Bull. Korean Math. Soc. 2023; 60(1): 83-91
https://doi.org/10.4134/BKMS.b210917
John Maxwell Campbell
Bull. Korean Math. Soc. 2023; 60(4): 1017-1024
https://doi.org/10.4134/BKMS.b220457
Xingyu Lei, Shuchao Li, Jianfeng Wang
Bull. Korean Math. Soc. 2023; 60(4): 873-893
https://doi.org/10.4134/BKMS.b220340
Junkee Jeon, Hyeng Keun Koo
Bull. Korean Math. Soc. 2023; 60(4): 1101-1129
https://doi.org/10.4134/BKMS.b220553
Karim Bouchannafa, Moulay Abdallah Idrissi, Lahcen Oukhtite
Bull. Korean Math. Soc. 2023; 60(5): 1281-1293
https://doi.org/10.4134/BKMS.b220654
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