JOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS Volume 14, Number 4, Winter 2002 INTEGRAL OPERATORS OF MARCINKIEWICZ TYPE AHMAD AL-SALMAN AND HUSSAIN AL-QASSEM ABSTRACT. In this paper we study integral operators of Marcinkiewicz type. We formulate a general method which allows us to obtain the L p boundedness of several classes of integral operators of Marcinkiewicz type. Our results extend as well as improve previously known results on Marcinkiewicz integral operators. 1. Introduction and statements of results. Let n ≥ 2 and S n-1 be the unit sphere in R n equipped with the normalized Lebesgue measure dσ. Suppose that Ω is a homogeneous function of degree zero on R n that satisfies Ω ∈ L 1 (S n-1 ) and (1.1)  S n-1 Ω(x) dσ(x)=0. Let U(r) be the open ball centered at the origin in R n with radius 2 r , r ∈ R. If r = ∞, we shall let U(r)= R n . For a suitable mapping Θ: U(r) → R d , d ∈ N and a measurable function h : R + → R, let {σ t,Θ,Ω,h,r : t ∈ R} be the family of measures defined on R d by (1.2)  R d fdσ t,Θ,Ω,h,r =2 -t χ (-∞,r) (t)  |y|≤2 t f (Θ(y))|y| 1-n Ω(y)h(|y|) dy, where χ (-∞,r) (t) is the characteristic function of the interval (-∞,r). Define the operator S Θ,Ω,h,r by (1.3) S Θ,Ω,h,r f (x)=  ∞ -∞ |σ t,Θ,Ω,h,r ∗ f (x)| 2 dt  1/2 . 2000 AMS Mathematics Subject Classification. Primary 42B20, Secondary 42B15, 42B25. Key words and phrases. Marcinkiewicz integral operators, Fourier transform, submanifolds of finite type, surfaces of revolution, real-analytic submanifolds, maximal functions, rough kernels. This paper is supported by Yarmouk University Research Council. Copyright c 2002 Rocky Mountain Mathematics Consortium 343