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Anisotropic Classes of Homogeneous Pseudodifferential Symbols
Studia Mathematica
  • Árpád Bényi, Western Washington University
  • Marcin Bownik
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We define homogeneous classes of x-dependent anisotropic symbols S˙,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)].

Citation Information
Árpád Bényi and Marcin Bownik. "Anisotropic Classes of Homogeneous Pseudodifferential Symbols" Studia Mathematica Vol. 200 Iss. 1 (2010) p. 41 - 66
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