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We define homogeneous classes of x-dependent anisotropic symbolṡ S 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-dependentdoi:10.4064/sm200-1-3 fatcat:n6oggbgcsng7rol4ba5mm56hz4