Hussain Al-Qassem

On certain estimates for Marcinkiewicz integrals and extrapolation

We obtainLp estimates for parametric Marcinkiewicz integrals associated to polynomial mappings and with rough kernels on the unit sphere as well as on the radial direction. These estimates will allow us to use an extrapolation argument to obtain some new and improved results on Marcinkiewicz integrals. Also, such estimates provide us with a unifying approach in dealing with Marcinkiewicz integrals when the kernel function ω belongs to the class of block spacesB q(0,α) (Sn-1) as well as when ω belongs to the classL(logL)α (Sn-1). Our conditions on the kernels are known to be the best possible in their respective classes.

Rough Marcinkiewicz integral operators

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.

On weighted inequalities for parametric Marcinkiewicz integrals

We establish a weighted boundedness of a parametric Marcinkiewicz integral operator if is allowed to be in the block space for some and satisfies a mild integrability condition. We apply this conclusion to obtain the weighted boundedness for a class of the parametric Marcinkiewicz integral operators and related to the Littlewood-Paley-function and the area integral, respectively. It is known that the condition is optimal for the boundedness of.

On the boundedness of maximal operators and singular operators with kernels in

We establish the-boundedness for a class of singular integral operators and a class of related maximal operators when their singular kernels are given by functions in.

Singular integrals along surfaces on product domains

In this paper, we study the mapping properties of singular integral operator along surfaces of revolution. We prove Lp bounds (1<p<∞) for such singular integral operators as well as for their corresponding maximal truncated singular integrals if the singular kernels are allowed to be in certain block spaces.

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

We study singular integral operators along subvarieties determined by flat curves and kernels in the Hardy space H 1 (S n−1). We prove that these operators are bounded on L p for all p ∈ (1, ∞). Our results extend previously known results.

Rough singular integrals on product spaces

We study the mapping properties of singular integral operators defined by mappings of finite type. We prove that such singular integral operators are bounded on the Lebesgue spaces under the condition that the singular kernels are allowed to be in certain block spaces.

Logarithmic Bounds for Oscillatory Singular Integrals on Hardy Spaces

We establish a logarithmic bound for oscillatory singular integrals with quadratic phases on the Hardy space . The logarithmic rate of growth is the best possible.

WEIGHTED ESTIMATES FOR ROUGH PARAMETRIC MARCINKIEWICZ INTEGRALS

We establish a weighted norm inequality for a class of rough parametric Marcinkiewicz integral operators MρΩ. As an application of this inequality, we obtain weighted Lp inequalities for a class of parametric Marcinkiewicz integral operators M∗,ρ Ω,λ and MρΩ,S related to the Littlewood-Paley g∗ λ-function and the area integral S, respectively.