Rough Marcinkiewicz integrals along certain smooth curves

Abstract

This paper is devoted to the study of the multiple-parameter rough Marcinkiewicz integral operators associated with certain smooth curves. It is shown that the Grafakos-Stefanov type size condition F (alpha) (S (m-1) x S (n-1)) of the kernel implies the L (p) -boundedness of these Marcinkiewicz integral operators for some alpha > 1/2 and , which is an essential improvement of certain previous results.National Natural Science Foundation of China [11071200]; Natural Science Foundation of Fujian Province [2010J01013]; Natural Science Foundation of Hunan Province [09JJ5002]; Natural Science Foundation of Zhejing Province [Y6100810

$L^2$ continuity of the Calderón type commutator for the Littlewood-Paley operator with rough variable kernel

“…W is a Littlewood-Paley function, that was studied by many authors (see [1,23,32,33,37,36,39]). For a general Wðx; z 0 Þ, Ding, Lin and Shao [25] gave the L 2 boundedness of m 1 W with variable kernel.…”

A rough Marcinkiewicz integral along smooth curves

“…For the sake of simplicity, we denote M ρ h,Ω,Φ,ϕ = M ρ h,Ω if Φ(t) = ϕ(t) = t and M ρ h,Ω = M ρ Ω if h(t) ≡ 1. When ρ ≡ 1, the operator M ρ Ω reduces to the classical Marcinkiewicz integral operator denoted by M Ω , which was introduced by Stein [20] and investigated by many authors (see [4,6,9,18,[20][21][22][23] for examples). In particular, Ding et al [9] (resp., Al-Salman et al [4]) showed that M Ω was bounded on L p (R n ) for 1 < p < ∞ provided that Ω ∈ H 1 (S n−1 ) (resp., Ω ∈ L(log + L) 1/2 (S n−1 )).…”