A class of generalized Marcinkiewicz integral operators is introduced, and, under rather weak conditions on the integral kernels, the boundedness of such operators on $L^p$ and Triebel--Lizorkin spaces is established.

In this paper, we give the $L^p$ boundedness for a class of parabolic Littlewood--Paley $g$-function with its kernel function $\Omega$ is in the Hardy space $H^1(S^{n-1})$.

In this paper, we study the $L^p$ mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on $L^p$ provided that their kernels satisfy a size condition much weaker than that for the classical Calder\'{o}n--Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions.

Let $b(t)$ be an $L^\infty$ function on $\bR$, $\Omega (\,y')$ be an $H^1$ function on the unit sphere satisfying the mean zero property (1) and $Q_m(t)$ be a real polynomial on $\bR$ of degree $m$ satisfying $Q_m(0)=0$. We prove that the singular integral operator $$ T_{Q_m,b} (\,f) (x)=p.v. \int_\bR^n b(|y|) \Omega(\,y) |y|^{-n} f \left( x-Q_m (|y|) y' \right) \,dy $$ is bounded in $L^p (\bR^n)$ for $1
Keywords:singular integral, rough kernel, Hardy space
Category:42B20