Mathematical Research Letters 9, 697–700 (2002) L p BOUNDS FOR THE FUNCTION OF MARCINKIEWICZ A. Al-Salman, H. Al-Qassem, L.C. Cheng, and Y. Pan 1. Introduction Let Ω denote a homogeneous function of degree 0 on R n which is locally integrable and satisfies  S n-1 Ω(y)dσ(y)=0, (1.1) where dσ represents the normalized Lebesgue measure on the unit sphere S n-1 . For n ≥ 2 and f ∈ L 1 loc (R n ), the Marcinkiewicz function of f is given by μ Ω (f )(x)=  ∞ 0      |y|≤t Ω(y) |y| n-1 f (x - y)dy     2 dt t 3  1/2 . (1.2) The above operator was introduced by E.M. Stein in [7] as an extension of the notion of Marcinkiewicz function from one dimension to higher dimensions. By using the L p boundedness of the 1-dimensional Marcinkiewicz function, Stein showed that μ Ω is bounded on L p (R n ) for 1 <p< ∞ whenever Ω is odd. For a general kernel function Ω, the L p boundedness of μ Ω has been estab- lished under various conditions on Ω. For example, Stein proved that μ Ω is bounded on L p (R n ) for 1 <p ≤ 2 if Ω ∈ Lip(S n-1 ). Benedek, Calder´ on and Panzone proved in [2] that the L p boundedness of μ Ω holds for 1 <p< ∞ under the condition that Ω ∈ C 1 (S n-1 ). In 1972 T. Walsh showed that the L p boundedness of μ Ω can still hold even if Ω is quite rough. Theorem 1 (Walsh [11]). Suppose that p ∈ (1, ∞),r = min{p, p ′ }, and Ω ∈ L(log L) 1/r (loglog L) 2(1-2/r ′ ) (S n-1 ). Then μ Ω is bounded on L p (R n ). When p = 2, the condition in Theorem 1 is simply Ω ∈ L(log L) 1/2 (S n-1 ), which was shown by Walsh to be optimal in the sense that the exponent 1/2 in L(log L) 1/2 cannot be replaced by any smaller numbers. On the other hand, Walsh did not consider his condition to be in any sense optimal when p = 2. Indeed, by comparing with the result of Calder´ on and Zygmund on singular integrals, one is naturally led to the question whether the condition Ω ∈ L(log L) 1/2 (S n-1 ) is also sufficient for the L p boundedness of μ Ω even when p = 2. This problem, which was formally proposed by Y. Ding in [4], is resolved by our next theorem. Received June 24, 2002. 697