We define homogeneous classes of x-dependent anisotropic symbols in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander-Mikhlin multipliers introduced by Rivière [Ark. Mat. 9 (1971)] and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by making use of the well-established Calderón-Zygmund theory on spaces of homogeneous type. We then show that x-dependent symbols in yield Calderón-Zygmund kernels, yet their L² boundedness fails. Finally, we prove boundedness results for the class on weighted anisotropic Besov and Triebel-Lizorkin spaces extending isotropic results of Grafakos and Torres [Michigan Math. J. 46 (1999)].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-1-3,
author = {\'Arp\'ad B\'enyi and Marcin Bownik},
title = {Anisotropic classes of homogeneous pseudodifferential symbols},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {41-66},
zbl = {1215.47038},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-1-3}
}
Árpád Bényi; Marcin Bownik. Anisotropic classes of homogeneous pseudodifferential symbols. Studia Mathematica, Tome 196 (2010) pp. 41-66. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-1-3/