Anisotropic classes of inhomogeneous pseudodifferential symbols

Abstract

We introduce a class of pseudodifferential operators in the anisotropic setting induced by an expansive dilation A which generalizes the classical isotropic class \({S^{m}_{\gamma, \delta}}\) of inhomogeneous symbols. We extend a well-known L 2-boundedness result to the anisotropic class \({S_{\delta, \delta}^0(A)}\), 0 ≤ δ < 1. As a consequence, we deduce that operators with symbols in the anisotropic class \({S^0_{1,0}(A)}\) are bounded on L p spaces, 1 < p < ∞.