Anisotropic classes of homogeneous pseudodifferential symbols

作者：Benyi Arpad^*; Bownik Marcin

来源：Studia Mathematica, 2010, 200(1): 41-66.

DOI：10.4064/sm200-1-3

• anisotropic symbol
• multiplier operator
• pseudodifferential operator
• Calderon-Zygmund operator
• space of homogeneous type

We define homogeneous classes of x-dependent anisotropic symbols S(over dot)(gamma,delta)(m)(A) in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hormander- Mikhlin multipliers introduced by Riviere [Ark. Mat. 9 (1971)] and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by making use of the well-established Calderon-Zygmund theory on spaces of homogeneous type. We then show that x-dependent symbols in S(over dot)(1,1)(0)(A) yield Calderon-Zygmund kernels, yet their L(2) boundedness fails. Finally, we prove boundedness results for the class S(over dot)(1,1)(m)(A) on weighted anisotropic Besov and Triebel-Lizorkin spaces extending isotropic results of Grafakos and Torres [Michigan Math. J. 46 (1999)].

• 出版日期2010