Anisotropic Classes of Homogeneous Pseudodifferential Symbols

We define homogeneous classes of x-dependent anisotropic symbols Ṡ 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-Mihlin multipliers introduced by Rivière [22] 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 xdependent symbols in Ṡ 1,1(A) exhibit Calderón-Zygmund kernels, yet their L boundedness fails. Finally, we prove boundedness results of the class Ṡ 1,1(A) on weighted anisotropic Besov and Triebel-Lizorkin spaces extending isotropic results of Grafakos and Torres [15].