In this paper we shall compare three notions of pointwise smoothness: the usual definition, J.M. Bony's two-microlocal spaces Cx0 s,s', and the corresponding definition on the wavelet coefficients. The purpose is mainly to show that these two-microlocal spaces provide "good substitutes" for the pointwise Hölder regularity condition; they can be very precisely compared with this condition, they have more functional properties, and can be characterized by conditions on the wavelet coefficients. We also give applications of these properties. In Part 2 some results on the microlocal spaces contained in [B2] will be recalled. Theorems 3 and 4 are also essentially contained in [B2]. The starting point for this paper was a note ([J1]) the author had written on a comparison between the Hölder criterion of regularity at a given point x0 and a corresponding property defined on the wavelet coefficients. Some easy proofs are omitted or abridged and can be found in [J2].
@article{urn:eudml:doc:41681,
title = {Pointwise smoothness, two-microlocalization and wavelet coefficients.},
journal = {Publicacions Matem\`atiques},
volume = {35},
year = {1991},
pages = {155-168},
mrnumber = {MR1103613},
zbl = {0760.42016},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41681}
}
Jaffard, Stéphane. Pointwise smoothness, two-microlocalization and wavelet coefficients.. Publicacions Matemàtiques, Tome 35 (1991) pp. 155-168. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41681/