Ahmad Al-Salman

Singular Integrals with Rough Kernels in L log L(Sn−1)

A systematic treatment is given of several classes of singular integrals. Their

A note on Marcinkiewicz integral operators

L^p

Rough Marcinkiewicz integral operators

boundedness is proved when their kernels are given by functions

On a class of singular integral operators with rough kernels

\Omega

Singular Integrals Along Flat Curves with Kernels in the Hardy Space H1(S n-1)

in

A unifying approach for certain class of maximal functions

L\log L({\bf S}^{n-1})

Rough singular integrals on product spaces

.

MARCINKIEWICZ INTEGRALS ALONG SUBVARIETIES ON PRODUCT DOMAINS

Abstract This paper is primarily concerned with proving the L p boundedness of Marcinkiewicz integral operators with kernels belonging to certain block spaces. We also show the optimality of our condition on the kernel for the L 2 boundedness of the Marcinkiewicz integral.

Singular Integrals With Rough Kernels

We study the Marcinkiewicz integral operator M𝒫f(x)=(∫−∞∞|∫|y|≤2tf(x−𝒫(y))(Ω(y)/|y|n−1)dy|2dt/22t)1/2, where 𝒫 is a polynomial mapping from ℝn into ℝd and Ω is a homogeneous function of degree zero on ℝn with mean value zero over the unit sphere Sn−1. We prove an Lp boundedness result of M𝒫 for rough Ω.

Marcinkiewicz integrals on product spaces

In this paper, we study the Lp 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 Lp provided that their kernels satisfy a size condition much weaker than that for the classical Calder´ on- Zygmund singular integral operators. Moreover, we present an example showing that our size condi- tion is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions.