Talks given at NuHAG events

Generalized 2-microlocal Spaces

  Henning Kempka

  given at  strobl07 (22.06.07 09:00)
  id:  572
  length:  25min
  status:  accepted
  type:  talk
\Large{\textbf{Generalized 2-microlocal Spaces}}\\[2ex]
\large{Henning Kempka (Jena)}\\
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
0 2^{-\alpha_1}w_j(x)&\leq
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
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
w_j(x)=(1+2^j|x-x_0|)^{s'}\quad\text{for some
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}.
\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)

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