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51. CMB 2007 (vol 50 pp. 3)

Basener, Richard F.
 Higher Dimensional Spaces of Functions on the Spectrum of a Uniform Algebra In this paper we introduce a nested family of spaces of continuous functions defined on the spectrum of a uniform algebra. The smallest space in the family is the uniform algebra itself. In the finite dimensional'' case, from some point on the spaces will be the space of all continuous complex-valued functions on the spectrum. These spaces are defined in terms of solutions to the nonlinear Cauchy--Riemann equations as introduced by the author in 1976, so they are not generally linear spaces of functions. However, these spaces do shed light on the higher dimensional properties of a uniform algebra. In particular, these spaces are directly related to the generalized Shilov boundary of the uniform algebra (as defined by the author and, independently, by Sibony in the early 1970s). Categories:32A99, 46J10

52. CMB 2006 (vol 49 pp. 508)

Cho, Hong Rae
 Growth Spaces and Growth Norm Estimates for $\Bar\partial$ on Convex Domains of Finite Type We consider the growth norm of a measurable function $f$ defined by $$\|f\|_{-\sigma}=\ess\{\delta_D(z)^\sigma|f(z)|:z\in D\},$$ where $\delta_D(z)$ denote the distance from $z$ to $\partial D$. We prove some optimal growth norm estimates for $\bar\partial$ on convex domains of finite type. Categories:32W05, 32A26, 32A36

53. CMB 2006 (vol 49 pp. 628)

Zeron, E. S.
 Approximation and the Topology of Rationally Convex Sets Considering a mapping $g$ holomorphic on a neighbourhood of a rationally convex set $K\subset\cc^n$, and range into the complex projective space $\cc\pp^m$, the main objective of this paper is to show that we can uniformly approximate $g$ on $K$ by rational mappings defined from $\cc^n$ into $\cc\pp^m$. We only need to ask that the second \v{C}ech cohomology group $\check{H}^2(K,\zz)$ vanishes. Keywords:Rationally convex, cohomology, homotopyCategories:32E30, 32Q55

54. CMB 2006 (vol 49 pp. 381)

Girela, Daniel; Peláez, José Ángel
 On the Membership in Bergman Spaces of the Derivative of a Blaschke Product With Zeros in a Stolz Domain It is known that the derivative of a Blaschke product whose zero sequence lies in a Stolz angle belongs to all the Bergman spaces $A^p$ with $01$). As a consequence, we prove that there exists a Blaschke product $B$ with zeros on a radius such that $B'\notin A^{3/2}$. Keywords:Blaschke products, Hardy spaces, Bergman spacesCategories:30D50, 30D55, 32A36

55. CMB 2006 (vol 49 pp. 237)

Gauthier, P. M.; Zeron, E. S.
 Approximation by Rational Mappings, via Homotopy Theory Continuous mappings defined from compact subsets $K$ of complex Euclidean space $\cc^n$ into complex projective space $\pp^m$ are approximated by rational mappings. The fundamental tool employed is homotopy theory. Keywords:Rational approximation, homotopy type, null-homotopicCategories:32E30, 32C18

56. CMB 2006 (vol 49 pp. 256)

Neelon, Tejinder
 A Bernstein--Walsh Type Inequality and Applications A Bernstein--Walsh type inequality for $C^{\infty }$ functions of several variables is derived, which then is applied to obtain analogs and generalizations of the following classical theorems: (1) Bochnak--Siciak theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real analytic on every line is real analytic; (2) Zorn--Lelong theorem: if a double power series $F(x,y)$\ converges on a set of lines of positive capacity then $F(x,y)$\ is convergent; (3) Abhyankar--Moh--Sathaye theorem: the transfinite diameter of the convergence set of a divergent series is zero. Keywords:Bernstein-Walsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power seriesCategories:32A05, 26E05

57. CMB 2006 (vol 49 pp. 72)

Dwilewicz, Roman J.
 Additive Riemann--Hilbert Problem in Line Bundles Over $\mathbb{CP}^1$ In this note we consider $\overline\partial$-problem in line bundles over complex projective space $\mathbb{CP}^1$ and prove that the equation can be solved for $(0,1)$ forms with compact support. As a consequence, any Cauchy-Riemann function on a compact real hypersurface in such line bundles is a jump of two holomorphic functions defined on the sides of the hypersurface. In particular, the results can be applied to $\mathbb{CP}^2$ since by removing a point from it we get a line bundle over $\mathbb{CP}^1$. Keywords:$\overline\partial$-problem, cohomology groups, line bundlesCategories:32F20, 14F05, 32C16

58. CMB 2005 (vol 48 pp. 601)

 On the Regularity of the $s$-Differential Metric We show that the injective Kobayashi--Royden differential metric, as defined by Hahn, is upper semicontinous. Keywords:Invariant metric, Kobayashi--Royden metric, Hahn metric, $s$-metricCategories:32F45, 32Q45

59. CMB 2005 (vol 48 pp. 500)

Baracco, Luca
 Extension of Holomorphic Functions From One Side of a Hypersurface We give a new proof of former results by G. Zampieri and the author on extension of holomorphic functions from one side $\Omega$ of a real hypersurface $M$ of $\mathbb{C}^n$ in the presence of an analytic disc tangent to $M$, attached to $\bar\Omega$ but not to $M$. Our method enables us to weaken the regularity assumptions both for the hypersurface and the disc. Keywords:analytic discs, Poisson integral, holomorphic extensionCategories:32D10, 32V25

60. CMB 2005 (vol 48 pp. 473)

Zeron, E. S.
 Logarithms and the Topology of the Complement of a Hypersurface This paper is devoted to analysing the relation between the logarithm of a non-constant holomorphic polynomial $Q(z)$ and the topology of the complement of the hypersurface defined by $Q(z)=0$. Keywords:Logarithm, homology groups and periodsCategories:32Q55, 14F45

61. CMB 2005 (vol 48 pp. 409)

Gauthier, P. M.; Xiao, J.
 The Existence of Universal Inner Functions on the Unit Ball of $\mathbb{C}^n$ It is shown that there exists an inner function $I$ defined on the unit ball ${\bf B}^n$ of ${\mathbb C}^n$ such that each function holomorphic on ${\bf B}^n$ and bounded by $1$ can be approximated by non-Euclidean translates" of $I$. Keywords:universal inner functionsCategories:32A35, 30D50, 47B38

62. CMB 2004 (vol 47 pp. 133)

Wang, Wei
 Embeddability of Some Three-Dimensional Weakly Pseudoconvex ${\rm CR}$ Structures We prove that a class of perturbations of standard ${\rm CR}$ structure on the boundary of three-dimensional complex ellipsoid $E_{p,q}$ can be realized as hypersurfaces on $\mathbb{C}^2$, which generalizes the result of Burns and Epstein on the embeddability of some perturbations of standard ${\rm CR}$ structure on $S^3$. Keywords:deformations, embeddability, complex ellipsoidsCategories:32V30, 32G07, 32V35

63. CMB 2003 (vol 46 pp. 559)

Marco, Nicolas; Massaneda, Xavier
 On Density Conditions for Interpolation in the Ball In this paper we study interpolating sequences for two related spaces of holomorphic functions in the unit ball of $\C^n$, $n>1$. We first give density conditions for a sequence to be interpolating for the class $A^{-\infty}$ of holomorphic functions with polynomial growth. The sufficient condition is formally identical to the characterizing condition in dimension $1$, whereas the necessary one goes along the lines of the results given by Li and Taylor for some spaces of entire functions. In the second part of the paper we show that a density condition, which for $n=1$ coincides with the characterizing condition given by Seip, is sufficient for interpolation in the (weighted) Bergman space. Categories:32A36, 32A38, 30E05

64. CMB 2003 (vol 46 pp. 321)

Ballico, E.
 Discreteness For the Set of Complex Structures On a Real Variety Let $X$, $Y$ be reduced and irreducible compact complex spaces and $S$ the set of all isomorphism classes of reduced and irreducible compact complex spaces $W$ such that $X\times Y \cong X\times W$. Here we prove that $S$ is at most countable. We apply this result to show that for every reduced and irreducible compact complex space $X$ the set $S(X)$ of all complex reduced compact complex spaces $W$ with $X\times X^\sigma \cong W\times W^\sigma$ (where $A^\sigma$ denotes the complex conjugate of any variety $A$) is at most countable. Categories:32J18, 14J99, 14P99

65. CMB 2003 (vol 46 pp. 429)

Sastry, Pramathanath; Tong, Yue Lin L.
 The Grothendieck Trace and the de Rham Integral On a smooth $n$-dimensional complete variety $X$ over ${\mathbb C}$ we show that the trace map ${\tilde\theta}_X \colon\break H^n (X,\Omega_X^n) \to {\mathbb C}$ arising from Lipman's version of Grothendieck duality in \cite{ast-117} agrees with $$(2\pi i)^{-n} (-1)^{n(n-1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb C}) \to {\mathbb C}$$ under the Dolbeault isomorphism. Categories:14F10, 32A25, 14A15, 14F05, 18E30

66. CMB 2003 (vol 46 pp. 291)

Sankaran, Parameswaran
 A Coincidence Theorem for Holomorphic Maps to $G/P$ The purpose of this note is to extend to an arbitrary generalized Hopf and Calabi-Eckmann manifold the following result of Kalyan Mukherjea: Let $V_n = \mathbb{S}^{2n+1} \times \mathbb{S}^{2n+1}$ denote a Calabi-Eckmann manifold. If $f,g \colon V_n \longrightarrow \mathbb{P}^n$ are any two holomorphic maps, at least one of them being non-constant, then there exists a coincidence: $f(x)=g(x)$ for some $x\in V_n$. Our proof involves a coincidence theorem for holomorphic maps to complex projective varieties of the form $G/P$ where $G$ is complex simple algebraic group and $P\subset G$ is a maximal parabolic subgroup, where one of the maps is dominant. Categories:32H02, 54M20

67. 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

68. CMB 2002 (vol 45 pp. 417)

Kamiyama, Yasuhiko; Tsukuda, Shuichi
 On Deformations of the Complex Structure on the Moduli Space of Spatial Polygons For an integer $n \geq 3$, let $M_n$ be the moduli space of spatial polygons with $n$ edges. We consider the case of odd $n$. Then $M_n$ is a Fano manifold of complex dimension $n-3$. Let $\Theta_{M_n}$ be the sheaf of germs of holomorphic sections of the tangent bundle $TM_n$. In this paper, we prove $H^q (M_n,\Theta_{M_n})=0$ for all $q \geq 0$ and all odd $n$. In particular, we see that the moduli space of deformations of the complex structure on $M_n$ consists of a point. Thus the complex structure on $M_n$ is locally rigid. Keywords:polygon space, complex structureCategories:14D20, 32C35

69. CMB 2002 (vol 45 pp. 80)

Gauthier, P. M.; Zeron, E. S.
 Approximation On Arcs and Dendrites Going to Infinity in $\C^n$ On a locally rectifiable arc going to infinity, each continuous function can be approximated by entire functions. Keywords:tangential approximation, CarlemanCategories:32E30, 32E25

70. CMB 2001 (vol 44 pp. 150)

Jakóbczak, Piotr
 Exceptional Sets of Slices for Functions From the Bergman Space in the Ball Let $B_N$ be the unit ball in $\mathbb{C}^N$ and let $f$ be a function holomorphic and $L^2$-integrable in $B_N$. Denote by $E(B_N,f)$ the set of all slices of the form $\Pi =L\cap B_N$, where $L$ is a complex one-dimensional subspace of $\mathbb{C}^N$, for which $f|_{\Pi}$ is not $L^2$-integrable (with respect to the Lebesgue measure on $L$). Call this set the exceptional set for $f$. We give a characterization of exceptional sets which are closed in the natural topology of slices. Categories:32A37, 32A22

71. CMB 2001 (vol 44 pp. 105)

Pilipović, Stevan
 Convolution Equation in $\mathcal{S}^{\prime\ast}$---Propagation of Singularities The singular spectrum of $u$ in a convolution equation $\mu * u = f$, where $\mu$ and $f$ are tempered ultradistributions of Beurling or Roumieau type is estimated by $$SS u \subset (\mathbf{R}^n \times \Char \mu) \cup SS f.$$ The same is done for $SS_{*}u$. Categories:32A40, 46F15, 58G07

72. CMB 2001 (vol 44 pp. 126)

Zeron, E. Santillan
 Each Copy of the Real Line in $\C^2$ is Removable Around 1995, Professors Lupacciolu, Chirka and Stout showed that a closed subset of $\C^N$ ($N\geq 2$) is removable for holomorphic functions, if its topological dimension is less than or equal to $N-2$. Besides, they asked whether closed subsets of $\C^2$ homeomorphic to the real line (the simplest 1-dimensional sets) are removable for holomorphic functions. In this paper we propose a positive answer to that question. Keywords:holomorphic function, removable setCategory:32D20

73. CMB 2000 (vol 43 pp. 294)

Bracci, Filippo
 Fixed Points of Commuting Holomorphic Maps Without Boundary Regularity We identify a class of domains of $\C^n$ in which any two commuting holomorphic self-maps have a common fixed point. Keywords:Holomorphic self-maps, commuting functions, fixed points, Wolff point, Julia's LemmaCategories:32A10, 32A40, 32H15, 32A30

74. CMB 2000 (vol 43 pp. 174)

Gantz, Christian; Steer, Brian
 Stable Parabolic Bundles over Elliptic Surfaces and over Riemann Surfaces We show that the use of orbifold bundles enables some questions to be reduced to the case of flat bundles. The identification of moduli spaces of certain parabolic bundles over elliptic surfaces is achieved using this method. Categories:14J27, 32L07, 14H60, 14D20

75. CMB 2000 (vol 43 pp. 47)

 A Property of Lie Group Orbits Let $G$ be a real Lie group and $X$ a real analytic manifold. Suppose that $G$ acts analytically on $X$ with finitely many orbits. Then the orbits are subanalytic in $X$. As a consequence we show that the micro-support of a $G$-equivariant sheaf on $X$ is contained in the conormal variety of the $G$-action. Categories:32B20, 22E15