# Generalized 2-microlocal Spaces

Henning Kempka

given at  strobl07 (22.06.07 09:00)
id:  572
length:  25min
status:  accepted
type:  talk
ABSTRACT:
\documentclass{scrartcl}
\usepackage[intlimits]{amsmath}
\usepackage{amssymb}
\begin{document}
\thispagestyle{empty}
\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 spacesC^{s,s'}_{x_0}(\mathbb{R}^n)$, as described in \cite{JaffardMeyer96}, are a special case of \eqref{1} with$p=q=\inftyand the weight functions \begin{align} w_j(x)=(1+2^j|x-x_0|)^{s'}\quad\text{for somex_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}
\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, Stï¿½phane: \emph{Pointwise smoothness, two-microlocalisation and wavelet coefficients}\\
Publications Mathematiques \textbf{35} (1991), 155-168
\bibitem{JaffardMeyer96}Jaffard, Stï¿½phane; 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: Birkhï¿½user (2006)
\end{thebibliography}}
\end{document}

Enter here the CODE for editing this talk:
If you have forgotten the CODE for your talk click here to send an email to the Webmaster!
NOTICE: In [EDIT-MODUS] you can also UPLOAD a presentation"