Anisotropic classes of homogeneous pseudodifferential symbols

• Autorzy: Árpád Bényi , Marcin Bownik
• Języki publikacji: 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)].

Kategorie tematyczne

• 42B15: Multipliers
• 42C40: Wavelets and other special systems
• 42B35: Function spaces arising in harmonic analysis
• 47G30: Pseudodifferential operators
• 43A85: Analysis on homogeneous spaces
• 42B20: Singular and oscillatory integrals (Calder\'on-Zygmund, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2010
• Tom: 200
• Numer: 1
• Strony: 41-66

Daty

wydano

2010

Twórcy

autor

Árpád Bényi

• Department of Mathematics, Western Washington University, 516 High Street, Bellingham, WA 98225-9063, U.S.A.

autor

Marcin Bownik

• 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

Identyfikatory

DOI

10.4064/sm200-1-3