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since the kernel of T , (sin x)/x, has a derivative which does not decay quickly enough at infinity to apply the usual theory (see Davis and Chang [3]). Our aim in this paper is to show a result on singular integrals which in fact does include operators defined with kernels such as (sin x)/x.Our result will be a variant of a classical result of Calderón and Zygmund [1]. Actually, the statement will resemble that of a theorem found in Stein's [5] treatment of the Calderón and Zygmund theory. In order to state our results succinctly, we first introduce a little terminology.We say that a function p ≥ 0 on R q satisfies a reverse-L ∞ inequality (abbreviated as "p satisfies RL ∞ ") if there is a constant C such that for every cube Q ⊆ R q centered at the origin we have 0 < p| Q ∞ ≤ Cp Q . Here and throughout, p| Q denotes the restriction of the function p to Q and p Q denotes the average of the function p on Q. With these notations and conventions, we can now state our results.

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