In this paper, essentially developing the Stein method, we prove the Kol-Ž . mogorov᎐Stein inequality for any Orlicz norm with the same constants . ᮊ 1996 Academic Press, Inc.
Abstract. Let 0 < p < oo, f(x) e Lp(U.n), and supp Ff be bounded, where F is the Fourier transform. We will prove in this paper that the sequence ll-DQ/]|i/|a'; a > o , has the same behavior as the sequence sup |£Q|'/I<«I; {esuppf/ a > 0. In other words, if we know all "far points" of supp Ff, we can wholly describe this behavior without any concrete calculation of ||Z)a/||p , a > 0 . A Paley-Wiener-Schwartz theorem for a nonconvex case, which is a consequence of the result, is given.
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