Anisotropic classes of homogeneous pseudodifferential symbols
Volume 200 / 2010
Abstract
We define homogeneous classes of $x$-dependent anisotropic symbols $\dot{S}^{m}_{\gamma, \delta}(A)$ 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 $\dot{S}^0_{1,1} (A)$ yield Calderón–Zygmund kernels, yet their $L^2$ boundedness fails. Finally, we prove boundedness results for the class $\dot{S}^m_{1,1} (A)$ on weighted anisotropic Besov and Triebel–Lizorkin spaces extending isotropic results of Grafakos and Torres [Michigan Math. J. 46 (1999)].
