A note on singular integrals with dominating mixed smoothness in Triebel-Lizorkin spaces

Let h   be a measurable function defined on ℝ+ × ℝ+. Let Ω ∈ L(log L+)vq(Sn1−1 × Sn2−1) (1 ≤ vq ≤ 2)ℝ+ × ℝ+. Let Ω ∈ L(log L+)vq(Sn1−1 × Sn2−1) (1 ≤ vq ≤ 2) be homogeneous of degree zero and satisfy certain cancellation conditions. We show that the singular integral Tf(x1,x2) = p. v. ∬ℝn1+n2Ω(y′1,y′2)h(|y1|,|y2|)|y1|n1|y2|n2f(x1−y1,x2−y2)dy1dy2maps from Sp,qα1,α2F˙(ℝn1 × ℝn2) boundedly to itself for 1 < p, q < ∞, α1, α2 ∈ ℝ.1 < p, q < ∞, α1, α2 ∈ ℝ.