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Introduction.In the present paper, we shall characterize some functions, those which satisfy a Lipschitz condition, as Fourier transforms of a certain sub-class of L p (R k ), and we shall give a contraction theorem of L p -Fourier transforms.A complex valued function f(x ly x 2 , , x k ) on R k , the k-άim. Euclidean space, is denoted by f(x).When / has the following property (i) or (ii), we say / is (p)normalized: for any finiteJi+y interval I, where 1/p + 1/p' = 1; (ii) if p = 1, then / is continuous and lim^,.^ f(x) = 0. We denote the j-th difference of /(a?), with respect to heR k , by Δl(f(x)), that is, Δί(f(x)) -Σ(-l) i+m ( 3 )f(x + mh) . m=0 \m)

We note that the conditions (~1) and (~) imply the following m j+l --1. n=mj We use K to denote a positive constant, which is not necessarily the same on any wo occurrences.)Mathematics subject classification numbers, 1991. Primary 42A28, 42A32.

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