Singular integrals with mixed homogeneity in Triebel–Lizorkin spaces

In this paper, we study the singular integral f (x) = p.v. Rn h(ρ(y))Ω(y ) ρβ (y) f (x− y)dy in Triebel–Lizorkin spaces. Here Ω ∈ L(log L+)(Sn−1) (n 2), h is a measurable, locally integrable function defined on (0,∞); and ρ is a norm which is homogeneous with respect to certain non-isotropic dilations. For the special case when ρ is the usual Euclidean norm, we also consider the singular integral above in weighted Triebel–Lizorkin spaces for some appropriate weights.