Anisotropic classes of homogeneous pseudodifferential symbols

Árpád Bényi, and Marcin Bownik. "Anisotropic classes of homogeneous pseudodifferential symbols." Studia Mathematica 200.1 (2010): 41-66. <http://eudml.org/doc/285876>.

@article{ÁrpádBényi2010,
abstract = {We define homogeneous classes of x-dependent anisotropic symbols $Ṡ^\{m\}_\{γ,δ\}(A)$ 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 $Ṡ⁰_\{1,1\}(A)$ yield Calderón-Zygmund kernels, yet their L² boundedness fails. Finally, we prove boundedness results for the class $Ṡ^m_\{1,1\}(A)$ on weighted anisotropic Besov and Triebel-Lizorkin spaces extending isotropic results of Grafakos and Torres [Michigan Math. J. 46 (1999)].},
author = {Árpád Bényi, Marcin Bownik},
journal = {Studia Mathematica},
keywords = {anisotropic; Besov spaces; Triebel-Lizorkin spaces; pseudodifferential operator; homogeneous symbol; Calderon-Zygmund operator},
language = {eng},
number = {1},
pages = {41-66},
title = {Anisotropic classes of homogeneous pseudodifferential symbols},
url = {http://eudml.org/doc/285876},
volume = {200},
year = {2010},
}

TY - JOUR
AU - Árpád Bényi
AU - Marcin Bownik
TI - Anisotropic classes of homogeneous pseudodifferential symbols
JO - Studia Mathematica
PY - 2010
VL - 200
IS - 1
SP - 41
EP - 66
AB - We define homogeneous classes of x-dependent anisotropic symbols $Ṡ^{m}_{γ,δ}(A)$ 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 $Ṡ⁰_{1,1}(A)$ yield Calderón-Zygmund kernels, yet their L² boundedness fails. Finally, we prove boundedness results for the class $Ṡ^m_{1,1}(A)$ on weighted anisotropic Besov and Triebel-Lizorkin spaces extending isotropic results of Grafakos and Torres [Michigan Math. J. 46 (1999)].
LA - eng
KW - anisotropic; Besov spaces; Triebel-Lizorkin spaces; pseudodifferential operator; homogeneous symbol; Calderon-Zygmund operator
UR - http://eudml.org/doc/285876
ER -