H1(T) is the space of integrable functions f on the circle T such that the Fourier coefficients f̂(n) vanish for negative integers n. A multiplier is by definition a map m of H1 to itself such that the Fourier transform diagonal izes m. Let m̂(n) denote the diagonal coefficients of m for nonnegative n. Then m is called idempotent if each coefficient is zero or one.
Theorem: If m is idempotent, then the set of n for which m̂(n) = 1 is a finite Boolean combination of sets of nonnegative integers of the following three types: finite sets, arithmetic sequences, and lacunary sequences.
By definition, a sequence is lacunary if there is a real number q > 1 such that each term of the sequence is at least as large as q times the preceding term. The theorem implies a classification of the projections in H1 which commute with translations, or, what is equivalent on the circle (but not on the line), of the closed, translation invariant subspaces which are complemented in H1. In the course of the proof, a 1 ower bound is obtai ned on the operator norm of a multiplier whose coefficients are 0 or greater than 1 in magnitude. This bound implies that the number of nonzero coefficients in disjoint intervals of the same length is the same, up to some factor depending on the norm of m, provided that both intervals are shorter than their distance from 0.
Part II is unrelated to Part I. There it is proved that a general expression measuring the oscillation of a function on an interval is minimized by the decreasing rearrangement of the function. A special case of this expression is the BMO norm for functions of bounded mean oscillation.
