In this paper, we study Marcinkiewicz integral operators with rough kernels supported by surfaces of revolutions. We prove that our operators are bounded on Lp under certain convexity assumptions on our surfaces and under very weak conditions on the kernel. | Turk J Math 26 (2002) , 329 – 338. ¨ ITAK ˙ c TUB Flat Marcinkiewicz Integral Operators Ahmad Al-Salman and Hussain Al-Qassem Abstract In this paper, we study Marcinkiewicz integral operators with rough kernels supported by surfaces of revolutions. We prove that our operators are bounded on Lp under certain convexity assumptions on our surfaces and under very weak conditions on the kernel. Key Words: Marcinkiewicz Integral, rough kernel, flat curves, Fourier transform. 1. Introduction Let Sn−1 be the unit sphere in Rn (n ≥ 2) equipped with the normalized Lebesgue measure dσ and Ω ∈ L1 Sn−1 be a homogeneous function of degree zero that satisfies Z Ω(x) dσ(x) = 0. () Sn−1 Let Γ : Rn → Rd , d ≥ n + 1 be a mapping such that the surface Γ(Rn ) is smooth in Rd . The Marcinkiewicz integral operator µΩ,Γ associated to Γ and Ω is defined by 2 Z 1 −n+1 f(x − Γ(y)) |y| Ω(y)dy 2−2tdt) 2 . t |y|≤2 −∞ Z µΩ,Γ f(x) = ( ∞ () The problem regarding the operator µΩ,Γ is that under what conditions on Γ and Ω, the operator µΩ,Γ maps Lp (Rd ) into Lp (Rd ) for some 1 1) properly and the condition Ω ∈ L(Log+ L) Sn−1 is known to be the most desirable on-Zygmund singular integral size condition for the Lp boundedness of the related Calder´ operator ([4]). We shall obtain Theorem as a consequence of a more general result in which we allow our kernels to be rough in the radial direction. To be more specific, for 1 0 330 Z R γ 1 |h(t)| dt) γ 1, let µφ,h be the operator defined by () with Γ(y) = (y, φ(|y|)) and Ω replaced by Ωh. Then we have the following theorem. Theorem . Suppose that φ : R+ → R is an increasing convex function and h ∈ ∆γ for some γ > 1. If Ω ∈ L(Log+ L) Sn−1 and satisfies (), then µφ,h is bounded on n o 0 Lp (Rn+1 ) for |1/p − 1/2| 0 Lemma . Suppose that h ∈ ∆γ for some γ > 1 and Ω ∈ L2 (Sn−1 ) with kΩkL1 ≤ 1. o n 0 Then for θ = min (3γ )−1 , (12)−1 , we have |ˆ σβ,φ,h (ξ, τ )| ≤ 2 khkγ kΩkL2 .
