STUDIA MATHEMATICA 190 (1) (2009) On rough maximal operators and Marcinkiewicz integrals along submanifolds by H. M. Al-Qassem (Doha) and Y. Pan (Pittsburgh, PA) Abstract. We investigate the L p boundedness for a class of parametric Marcinkiewicz integral operators associated to submanifolds and a class of related maximal operators under the L(log L) α (S n−1 ) condition on the kernel functions. Our results improve and extend some known results. 1. Introduction and statement of results. Let R n (n ≥ 2) be the n-dimensional Euclidean space and S n−1 be the unit sphere in R n equipped with the induced Lebesgue measure dσ = dσ(·). For x ∈ R n \{0}, let x ′ = x/|x|. Let Ω be a function in L 1 (S n−1 ) satisfying (1.1)  S n−1 Ω(x) dσ(x)=0. For 1 ≤ γ ≤∞, let ∆ γ (R + ) denote the collection of all measurable functions h : [0, ∞) → C satisfying sup R>0 (R −1  R 0 |h(t)| γ dt) 1/γ < ∞. We note that L ∞ (R + ) ⊂ ∆ β (R + ) ⊂ ∆ α (R + ) for α<β, L γ (R + ,dt/t) ⊂ ∆ γ (R + ) for 1 ≤ γ< ∞, and all these inclusions are proper. Let L(log L) α (S n−1 ) (for α> 0) denote the class of all measurable functions Ω on S n−1 which satisfy ‖Ω‖ L(log L) α (S n−1 ) =  S n−1 |Ω(y)| log α (2 + |Ω(y)|) dσ(y) < ∞. In this paper, we are interested in parametric Marcinkiewicz integral operators of the form 2000 Mathematics Subject Classification : Primary 42B20; Secondary 42B15, 42B25. Key words and phrases : Marcinkiewicz integrals, maximal operators, L p boundedness, rough kernel. The work on this paper was done while the first author was on sabbatical leave from Yarmouk University. [73] c  Instytut Matematyczny PAN, 2009