Document Type
Article
Publication Date
2010
Keywords
Anisotropic symbol, Multiplier operator, Space of homogeneous type
Abstract
We define homogeneous classes of x-dependent anisotropic symbols S˙mγ,δ(A) 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 S˙01,1(A) yield Calderón–Zygmund kernels, yet their L2 boundedness fails. Finally, we prove boundedness results for the class S˙m1,1(A)on weighted anisotropic Besov and Triebel–Lizorkin spaces extending isotropic results of Grafakos and Torres [Michigan Math. J. 46 (1999)].
Publication Title
Studia Mathematica
Volume
200
Issue
1
First Page
41
Last Page
66
Required Publisher's Statement
INSTYTUT MATEMATYCZNY · POLSKA AKADEMIA NAUK
http://journals.impan.gov.pl/sm/
Recommended Citation
Bényi, Árpád and Bownik, Marcin, "Anisotropic Classes of Homogeneous Pseudodifferential Symbols" (2010). Mathematics Faculty Publications. 48.
https://cedar.wwu.edu/math_facpubs/48
Subjects - Topical (LCSH)
Pseudodifferential operators; Calderón-Zygmund operator; Homogeneous spaces
Genre/Form
articles
Type
Text
Rights
Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.
Language
English
Format
application/pdf
