Singular integrals with mixed homogeneity in Triebel–Lizorkin spaces

In this paper, we study the singular integralℑf(x)=p.v.∫Rnh(ρ(y))Ω(y′)ρβ(y)f(x−y)dy in Triebel–Lizorkin spaces. Here Ω∈L(logL+)(Sn−1)Ω∈L(logL+)(Sn−1) (n⩾2n⩾2), h   is a measurable, locally integrable function defined on (0,∞)(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.