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1. CMB Online first
New Characterizations of the Weighted Composition Operators Between Bloch Type Spaces in the Polydisk |
New Characterizations of the Weighted Composition Operators Between Bloch Type Spaces in the Polydisk We give some new characterizations for compactness of weighted
composition operators $uC_\varphi$ acting on Bloch-type spaces in
terms of the power of the components of $\varphi,$ where $\varphi$
is a holomorphic self-map of the polydisk $\mathbb{D}^n,$ thus
generalizing the results obtained by HyvÃ¤rinen and
LindstrÃ¶m in 2012.
Keywords:weighted composition operator, compactness, Bloch type spaces, polydisk, several complex variables Categories:47B38, 47B33, 32A37, 45P05, 47G10 |
2. CMB 2009 (vol 53 pp. 11)
Approximation and Interpolation by Entire Functions of Several Variables Let $f\colon \mathbb R^n\to \mathbb R$ be $C^\infty$ and let $h\colon
\mathbb R^n\to\mathbb R$ be positive
and continuous. For any unbounded nondecreasing sequence $\{c_k\}$
of nonnegative real numbers and for any sequence without
accumulation points $\{x_m\}$ in $\mathbb R^n$, there exists an entire
function $g\colon\mathbb C^n\to\mathbb C$ taking real values on $\mathbb R^n$ such that
\begin{align*}
&|g^{(\alpha)}(x)-f^{(\alpha)}(x)|\lt h(x), \quad |x|\ge c_k, |\alpha|\le k,
k=0,1,2,\dots,
\\
&g^{(\alpha)}(x_m)=f^{(\alpha)}(x_m), \quad |x_m|\ge c_k, |\alpha|\le k,
m,k=0,1,2,\dots.
\end{align*}
This is a version for functions of several variables of the
case $n=1$ due to L. Hoischen.
Keywords:entire function, complex approximation, interpolation, several complex variables Category:32A15 |
3. CMB 2005 (vol 48 pp. 622)
Hyperplanes of the Form ${f_1(x,y)z_1+\dots+f_k(x,y)z_k+g(x,y)}$ Are Variables The Abhyankar--Sathaye Embedded Hyperplane Problem asks whe\-ther any
hypersurface of $\C^n$ isomorphic to $\C^{n-1}$ is rectifiable, {\em
i.e.,}
equivalent to a linear hyperplane up to an automorphism of $\C^n$.
Generalizing the approach adopted by Kaliman, V\'en\'ereau, and
Zaidenberg which
consists in using almost nothing but the acyclicity of $\C^{n-1}$, we solve
this problem for hypersurfaces given by polynomials of $\C[x,y,z_1,\dots, z_k]$
as in the title.
Keywords:variables, Abhyankar--Sathaye Embedding Problem Categories:14R10, 14R25 |