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76. CMB 1999 (vol 42 pp. 499)

Zaharia, Alexandru
 Characterizations of Simple Isolated Line Singularities A line singularity is a function germ $f\colon(\CC ^{n+1},0) \lra\CC$ with a smooth $1$-dimensional critical set $\Sigma=\{(x,y)\in \CC\times \CC^n \mid y=0\}$. An isolated line singularity is defined by the condition that for every $x \neq 0$, the germ of $f$ at $(x,0)$ is equivalent to $y_1^2 +\cdots+y_n ^2$. Simple isolated line singularities were classified by Dirk Siersma and are analogous of the famous $A-D-E$ singularities. We give two new characterizations of simple isolated line singularities. Categories:32S25, 14B05

77. CMB 1999 (vol 42 pp. 97)

Kwon, E. G.
 On Analytic Functions of Bergman $\BMO$ in the Ball Let $B = B_n$ be the open unit ball of $\bbd C^n$ with volume measure $\nu$, $U = B_1$ and ${\cal B}$ be the Bloch space on $U$. ${\cal A}^{2, \alpha} (B)$, $1 \leq \alpha < \infty$, is defined as the set of holomorphic $f\colon B \rightarrow \bbd C$ for which $$\int_B \vert f(z) \vert^2 \left( \frac 1{\vert z\vert} \log \frac 1{1 - \vert z\vert } \right)^{-\alpha} \frac {d\nu (z)}{1-\vert z\vert} < \infty$$ if $0 < \alpha <\infty$ and ${\cal A}^{2, 1} (B) = H^2(B)$, the Hardy space. Our objective of this note is to characterize, in terms of the Bergman distance, those holomorphic $f\colon B \rightarrow U$ for which the composition operator $C_f \colon {\cal B} \rightarrow {\cal A}^{2, \alpha}(B)$ defined by $C_f (g) = g\circ f$, $g \in {\cal B}$, is bounded. Our result has a corollary that characterize the set of analytic functions of bounded mean oscillation with respect to the Bergman metric. Keywords:Bergman distance, \BMOA$, Hardy space, Bloch functionCategory:32A37 78. CMB 1998 (vol 41 pp. 129) Lee, Young Joo  Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces A class of Toeplitz type operators acting on the weighted Bergman spaces of the unit ball in the$n$-dimensional complex space is considered and two pluriharmonic symbols of commuting Toeplitz type operators are completely characterized. Keywords:Pluriharmonic functions, Weighted Bergman spaces, Toeplitz type operators.Categories:47B38, 32A37 79. CMB 1997 (vol 40 pp. 356) Mazet, Pierre  Principe du maximum et lemme de Schwarz, a valeurs vectorielles Nous {\'e}tablissons un th{\'e}or{\e}me pour les fonctions holomorphes {\a} valeurs dans une partie convexe ferm{\'e}e. Ce th{\'e}or{\e}me pr{\'e}cise la position des coefficients de Taylor de telles fonctions et peut {\^e}tre consid{\'e}r{\'e} comme une g{\'e}n{\'e}ralisation des in{\'e}galit{\'e}s de Cauchy. Nous montrons alors comment ce th{\'e}or{\e}me permet de retrouver des versions connues du principe du maximum et d'obtenir de nouveaux r{\'e}sultats sur les applications holomorphes {\`a} valeurs vectorielles. Keywords:Principe du maximum, lemme de Schwarz, points extr{Ã©maux.Categories:30C80, 32A30, 46G20, 52A07 80. CMB 1997 (vol 40 pp. 129) Badea, Catalin  Sur les caractÃ¨res d'une algÃ¨bre de Banach A new proof for the Gleason-Kahane-\.Zelazko theorem concerning the characters of a Banach algebra is given. A theorem due to P\'olya and Saxer is used instead of the Hadamard factorization theorem. Categories:46H05, 32A15 81. CMB 1997 (vol 40 pp. 117) Vigué, Jean-Pierre  Un lemme de Schwarz pour les boules-unitÃ©s ouvertes Let$B_1$and$B_2$be the open unit balls of${\bbd C}^{n_1}$and${\bbd C}^{n_2}$for the norms$\Vert\,{.}\,\Vert_1$and$\Vert\,{.}\, \Vert_2$. Let$f \colon B_1 \rightarrow B_2$be a holomorphic mapping such that$f(0)=0$. It is well known that, for every$z \in B_1$,$\Vert f(z)\Vert_2 \leq \Vert z \Vert_1$, and$\Vert f'(0)\Vert \leq 1$. In this paper, I prove the converse of this result. Let$f \colon B_1 \rightarrow B_2$be a holomorphic mapping such that$f'(0)$is an isometry. If$B_2$is strictly convex, I prove that$f(0) =0$and that$f$is linear. I also define the rank of a point$x$belonging to the boundary of$B_1$or$B_2$. Under some hypotheses on the ranks, I prove that a holomorphic mapping such that$f(0) = 0$and that$f'(0)\$ is an isometry is linear. Categories:32H15, 32H02
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