Parabolic Littlewood-Paley g-Function with Rough Kernel

In this note, we give the L^p （1 〈 p 〈∞） boundedness of the parabolic Littlewood Paley g-function with rough kernel.

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

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 Th,Ω,φ be the classical singular Radon transform, and let Th^ε,Ω,φ be its truncated operator with rough kernels associated to polynomial mapping which is defined by Th^ε,Ω,φf(x)=|f|y|>εf(x-φ(y))h(|y|)Ω(y)|y|^-ndy|.In this paper, we show that for any a ∈(-∞,∞) and (p, q) satisfying certain index condition, the operator Th^ε,Ω,φ enjoys the following convergence properties lim ε→0||Th^ε,Ω,φf-Th,Ω,φf||Fα^p,q(R^d)= 0 and limε→0||Th^ε,Ω,φf-Th,Ω,φf||Bα^p,q(R^d)=0, provided that Ω∈L(log+ L)β(S^n-1) for some β∈(0,1], or Ω∈H^1(S^n-1), or Ω∈(U1<q<∞Bq^(0,0)(S^n-1)).

Boundedness and continuity of maximal singular integrals and maximal functions on Triebel-Lizorkin spaces

In this paper,we systematically study several classes of maximal singular integrals and maximal functions with rough kernels in Fβ(S^n-1),a topic that relates to the Grafakos-Stefanov class.The boundedness and continuity of these operators on Triebel-Lizorkin spaces and Besov spaces are discussed.