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Generalized 2-microlocal Spaces


  Henning Kempka

  given at  strobl07 (22.06.07 09:00)
  id:  572
  length:  25min
  status:  accepted
  type:  talk
  LINK-Preprint:  http://www.minet.uni-jena.de/Math-Net/reports/sources/2006/06-23report.pdf
  LINK-Presentation: 
  ABSTRACT:
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\begin{center}
\Large{\textbf{Generalized 2-microlocal Spaces}}\\[2ex]
\large{Henning Kempka (Jena)}\\
\scriptsize{\emph{khenning@minet.uni-jena.de}}
\end{center}
We give a generalization of 2-microlocal spaces in the sense of weighted Besov spaces.
The concept of 2-microlocal analysis or 2-microlocal function spaces
is due to J.M. Bony (see \cite{Bony}). It is an appropriate
instrument to describe the local regularity and the oscillatory
behavior of functions near to singularities.\\
The approach is Fourier-analytical using Littlewood-Paley-analysis
of distributions. The theory has been elaborated and widely used in
fractal analysis and signal processing by several authors. We refer
to \cite{Jaffard91}, \cite{JaffardMeyer96}, \cite{VehelSeuret04},
\cite{Meyer97} and \cite{Moritoh}.\\
Therefore, let $\{\varphi_j\}_{j\in\mathbb{N}_0}$ be a smooth resolution of
unity and let $\{w_j\}_{j\in\mathbb{N}_0}$ be a sequence of weight functions
satisfying
\begin{align}
0 2^{-\alpha_1}w_j(x)&\leq
w_{j+1}(x)\leq2^{\alpha_2}w_j(x),
\end{align}
for $x,y\in\mathbb{R}^n$, $j\in\mathbb{N}_0$ and $\alpha,\alpha_1,\alpha_2\geq0$.
Let $0 of all $f\in S'(\mathbb{R}^n)$ such that
\begin{align}\label{1}
\|f|B^{s,mloc}_{pq}(\mathbb{R}^n)\|=\left(\sum_{j=0}^\infty2^{jsq}\|w_j\mathcal{F}^{-1}(\varphi_j\mathcal{F}
f)|L_p(\mathbb{R}^n)\|^q\right)^{1/q}<\infty.
\end{align}
The usual 2-microlocal spaces $C^{s,s'}_{x_0}(\mathbb{R}^n)$, as described
in \cite{JaffardMeyer96}, are a
special case of \eqref{1} with $p=q=\infty$ and the weight functions
\begin{align}
w_j(x)=(1+2^j|x-x_0|)^{s'}\quad\text{for some
$x_0\in\mathbb{R}^n$.}
\end{align}
We give first properties of these spaces and and a
characterization in sequence spaces by wavelets. We follow
closely the ideas expressed in \cite{Triebel3}.
\scriptsize{
\begin{thebibliography}{99}
\addcontentsline{toc}{chapter}{References}
\bibitem{Bony}Bony, Jean-Michel: \emph{Second Microlocalization and Propagation of Singularities for Semi-Linear Hyperbolic Equations}\\
Taniguchi Symp. HERT. Katata (1984), 11-49
\bibitem{Jaffard91}Jaffard, St�phane: \emph{Pointwise smoothness, two-microlocalisation and wavelet coefficients}\\
Publications Mathematiques \textbf{35} (1991), 155-168
\bibitem{JaffardMeyer96}Jaffard, St�phane; Meyer, Yves: \emph{Wavelet methods for pointwise regularity and local oscillations of functions}\\
Memoirs of the AMS, vol. \textbf{123} (1996)
\bibitem{VehelSeuret04}L\'{e}vy Vehel, Jacques; Seuret, St\'{e}phane: \emph{The 2-Microlocal Formalism}\\
Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Proceedings of Symposia in Pure Mathematics, PSPUM, vol. \textbf{72}, part2
(2004), 153-215
\bibitem{Meyer97}Meyer, Yves: \emph{Wavelets, Vibrations and Scalings}\\
CRM monograph series, AMS, vol. \textbf{9} (1997)
\bibitem{Moritoh}Moritoh, Shinya; Yamada, Tomomi: \emph{Two-microlocal Besov spaces and wavelets}\\
Rev. Mat. Iberoamericana \textbf{20} (2004), 277-283
\bibitem{Triebel3}Triebel, Hans: \emph{Theory of Function Spaces III}\\
Basel: Birkh�user (2006)
\end{thebibliography}}
\end{document}


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