We define homogeneous classes of x-dependent anisotropic symbols S˙mγ,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)].
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INSTYTUT MATEMATYCZNY · POLSKA AKADEMIA NAUK
Bényi, Árpád and Bownik, Marcin, "Anisotropic Classes of Homogeneous Pseudodifferential Symbols" (2010). Mathematics. 48.