A class of nonhomogeneous singular integrals in Rn

Let {δt }t>0 be a nonisotropic dilation group on Rn, let ρ be a distance function on Rn which is homogeneous with respect to {δt }t>0, and for f ∈ C∞0 (Rn) define Tf = [p.v. ρ(·)−αei/|·|β] ∗ f , where α and β are positive parameters. We give necessary and sufficient conditions on p, α and β for which T extends to a bounded linear operator on Lp(Rn).  2003 Elsevier Inc. All rights reserved