Convergence of truncated rough singular integrals supported by subvarieties on Triebel-Lizorkin spaces

Abstract

Let Ω be a function of homogeneous of degree zero and satisfy the cancellation condition on the unit sphere. Suppose that h is a radial function. Let be the classical singular Radon transform, and let be its truncated operator with rough kernels associated to polynomial mapping , which is defined by . In this paper, we show that for any α ∈ (−∞, ∞) and (p, q) satisfying certain index condition, the operator enjoys the following convergence properties and , provided that Ω ∈ L(log+L)β(Sn−1) for some β ∈ ( 0, 1], or Ω ∈ H1(Sn−1), or $$\Omega\in(\cup_{1<q<\infty}B_q^{(0,0)}(S^{n-1}))$$ .