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• # Artykuł - szczegóły

## Studia Mathematica

2010 | 200 | 1 | 41-66

## Anisotropic classes of homogeneous pseudodifferential symbols

EN

### Abstrakty

EN
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)].

41-66

wydano
2010

### Twórcy

autor
• Department of Mathematics, Western Washington University, 516 High Street, Bellingham, WA 98225-9063, U.S.A.
autor
• Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, U.S.A.
• Institute of Mathematics, Polish Academy of Sciences, Abrahama 18, 81-825 Sopot, Poland