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Search: All articles in the CMB digital archive with keyword harmonic

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1. CMB 2011 (vol 55 pp. 597)

 Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales We determine the best constants $C_{p,\infty}$ and $C_{1,p}$, $1 < p < \infty$, for which the following holds. If $u$, $v$ are orthogonal harmonic functions on a Euclidean domain such that $v$ is differentially subordinate to $u$, then $$\|v\|_p \leq C_{p,\infty} \|u\|_\infty,\quad \|v\|_1 \leq C_{1,p} \|u\|_p.$$ In particular, the inequalities are still sharp for the conjugate harmonic functions on the unit disc of $\mathbb R^2$. Sharp probabilistic versions of these estimates are also studied. As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund. Keywords: harmonic function, conjugate harmonic functions, orthogonal harmonic functions, martingale, orthogonal martingales, norm inequality, optimal stopping problemCategories:31B05, 60G44, 60G40

2. CMB 2011 (vol 55 pp. 242)

Cegrell, Urban
 Convergence in Capacity In this note we study the convergence of sequences of Monge-AmpÃ¨re measures $\{(dd^cu_s)^n\}$, where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity. Keywords:complex Monge-AmpÃ¨re operator, convergence in capacity, plurisubharmonic functionCategories:32U20, 31C15

3. CMB 2011 (vol 55 pp. 88)

Ghanbari, K.; Shekarbeigi, B.
 Inequalities for Eigenvalues of a General Clamped Plate Problem Let $D$ be a connected bounded domain in $\mathbb{R}^n$. Let $0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues of the following Dirichlet problem: $$\begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in D u|_{\partial D}=\frac{\partial u}{\partial n}|_{\partial D}=0, \end{cases}$$ where $V(x)$ is a nonnegative potential, and $\rho(x)\in C(\bar{D})$ is positive. We prove the following inequalities: $$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}} {\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times \frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}-\mu_i)]^{1/2},$$ $$\frac{n^2k^2}{8(n+2)}\leq \Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}-\mu_i}\Bigr] \times\sum_{i=1}^k\mu_i^{1/2}.$$ Keywords:biharmonic operator, eigenvalue, eigenvector, inequalityCategory:35P15

4. CMB 2009 (vol 52 pp. 18)

Chinea, Domingo
 Harmonicity of Holomorphic Maps Between Almost Hermitian Manifolds In this paper we study holomorphic maps between almost Hermitian manifolds. We obtain a new criterion for the harmonicity of such holomorphic maps, and we deduce some applications to horizontally conformal holomorphic submersions. Keywords:almost Hermitian manifolds, harmonic maps, harmonic morphismCategories:53C15, 58E20

5. CMB 2008 (vol 51 pp. 448)

Sasahara, Toru
 Stability of Biharmonic Legendrian Submanifolds in Sasakian Space Forms Biharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed. Keywords:biharmonic maps, Sasakian manifolds, Legendrian submanifoldsCategories:53C42, 53C40

6. CMB 2007 (vol 50 pp. 113)

Li, ZhenYang; Zhang, Xi
 Hermitian Harmonic Maps into Convex Balls In this paper, we consider Hermitian harmonic maps from Hermitian manifolds into convex balls. We prove that there exist no non-trivial Hermitian harmonic maps from closed Hermitian manifolds into convex balls, and we use the heat flow method to solve the Dirichlet problem for Hermitian harmonic maps when the domain is a compact Hermitian manifold with non-empty boundary. Keywords:Hermitian harmonic map, Hermitian manifold, convex ballCategories:58E15, 53C07

7. CMB 2004 (vol 47 pp. 481)

Bekjan, Turdebek N.
 A New Characterization of Hardy Martingale Cotype Space We give a new characterization of Hardy martingale cotype property of complex quasi-Banach space by using the existence of a kind of plurisubharmonic functions. We also characterize the best constants of Hardy martingale inequalities with values in the complex quasi-Banach space. Keywords:Hardy martingale, Hardy martingale cotype,, plurisubharmonic functionCategories:46B20, 52A07, 60G44

8. CMB 2003 (vol 46 pp. 373)

Laugesen, Richard S.; Pritsker, Igor E.
 Potential Theory of the Farthest-Point Distance Function We study the farthest-point distance function, which measures the distance from $z \in \mathbb{C}$ to the farthest point or points of a given compact set $E$ in the plane. The logarithm of this distance is subharmonic as a function of $z$, and equals the logarithmic potential of a unique probability measure with unbounded support. This measure $\sigma_E$ has many interesting properties that reflect the topology and geometry of the compact set $E$. We prove $\sigma_E(E) \leq \frac12$ for polygons inscribed in a circle, with equality if and only if $E$ is a regular $n$-gon for some odd $n$. Also we show $\sigma_E(E) = \frac12$ for smooth convex sets of constant width. We conjecture $\sigma_E(E) \leq \frac12$ for all~$E$. Keywords:distance function, farthest points, subharmonic function, representing measure, convex bodies of constant widthCategories:31A05, 52A10, 52A40

9. CMB 2003 (vol 46 pp. 268)

Puls, Michael J.
 Group Cohomology and $L^p$-Cohomology of Finitely Generated Groups Let $G$ be a finitely generated, infinite group, let $p>1$, and let $L^p(G)$ denote the Banach space $\{ \sum_{x\in G} a_xx \mid \sum_{x\in G} |a_x |^p < \infty \}$. In this paper we will study the first cohomology group of $G$ with coefficients in $L^p(G)$, and the first reduced $L^p$-cohomology space of $G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups. Keywords:group cohomology, $L^p$-cohomology, central element of infinite order, harmonic function, continuous linear functionalCategories:43A15, 20F65, 20F18

10. CMB 2003 (vol 46 pp. 113)

Lee, Jaesung; Rim, Kyung Soo
 Properties of the $\mathcal{M}$-Harmonic Conjugate Operator We define the $\mathcal{M}$-harmonic conjugate operator $K$ and prove that it is bounded on the nonisotropic Lipschitz space and on $\BMO$. Then we show $K$ maps Dini functions into the space of continuous functions on the unit sphere. We also prove the boundedness and compactness properties of $\mathcal{M}$-harmonic conjugate operator with $L^p$ symbol. Keywords:$\mathcal{M}$-harmonic conjugate operatorCategories:32A70, 47G10

11. CMB 2002 (vol 45 pp. 154)

Weitsman, Allen
 On the Poisson Integral of Step Functions and Minimal Surfaces Applications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image. Keywords:harmonic mappings, dilatation, minimal surfacesCategories:30C62, 31A05, 31A20, 49Q05

12. CMB 2001 (vol 44 pp. 376)

Zhang, Xi
 A Note on $p$-Harmonic $1$-Forms on Complete Manifolds In this paper we prove that there is no nontrivial $L^{q}$-integrably $p$-harmonic $1$-form on a complete manifold with nonnegatively Ricci curvature $(0 Keywords:$p$-harmonic,$1$-form, complete manifold, Sobolev inequalityCategories:58E20, 53C21 13. CMB 2000 (vol 43 pp. 496) Xu, Yuan  Harmonic Polynomials Associated With Reflection Groups We extend Maxwell's representation of harmonic polynomials to$h$-harmonics associated to a reflection invariant weight function$h_k$. Let$\CD_i$,$1\le i \le d$, be Dunkl's operators associated with a reflection group. For any homogeneous polynomial$P$of degree$n$, we prove the polynomial$|\xb|^{2 \gamma +d-2+2n}P(\CD)\{1/|\xb|^{2 \gamma +d-2}\}$is a$h$-harmonic polynomial of degree$n$, where$\gamma = \sum k_i$and$\CD=(\CD_1,\ldots,\CD_d)$. The construction yields a basis for$h$-harmonics. We also discuss self-adjoint operators acting on the space of$h$-harmonics. Keywords:$h$-harmonics, reflection group, Dunkl's operatorsCategories:33C50, 33C45 14. CMB 1998 (vol 41 pp. 129) Lee, Young Joo  Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces A class of Toeplitz type operators acting on the weighted Bergman spaces of the unit ball in the$n$-dimensional complex space is considered and two pluriharmonic symbols of commuting Toeplitz type operators are completely characterized. Keywords:Pluriharmonic functions, Weighted Bergman spaces, Toeplitz type operators.Categories:47B38, 32A37 15. CMB 1997 (vol 40 pp. 60) Khavinson, Dmitry  Cauchy's problem for harmonic functions with entire data on a sphere We give an elementary potential-theoretic proof of a theorem of G.~Johnsson: all solutions of Cauchy's problems for the Laplace equations with an entire data on a sphere extend harmonically to the whole space${\bf R}^N\$ except, perhaps, for the center of the sphere. Keywords:harmonic functions, Cauchy's problem, homogeneous harmonicsCategories:35B60, 31B20