On Composition Operators Which Preserve Bmo

Q |f − fQ|dx < ∞, (1) where |Q| is the n-dimensional Lebesgue measure of Q, fQ = |Q|−1 ∫ Q fdx, and the supremum is taken over all closed cubes Q ⊂ D with sides parallel to the coordinate axes. Let D and D′ be subdomains of Rm and Rn, m, n ≥ 1, respectively. We say that a map F : D → D′ is measurable if F−1(E) is measurable for each measurable subset E of D′. We say that a measurable map F : D → D′ is a BMO map if i) for each null set E ⊂ D′, F−1(E) is also a null set, and furthermore, ii) for each BMO(D′) function f , CF (f) = f ◦ F belongs to BMO(D). The condition i) guarantees the uniqueness of the function f ◦F . From the closed graph theorem each BMO map F induces a bounded operator CF between BMO spaces. Various partial results are known for the characterization of BMO maps. It seems, however, that we do not know almost anything yet for noncontinuous BMO maps. The main purpose of the present paper is to give a characterization of BMO maps F : Rm → Rn, m,n ≥ 1 (Theorem 3.1). Our argument depends on the following two celebrated results for BMO; a growth estimation for BMO functions due to John-Nirenberg, and the existence of certain extremal BMO functions due to Uchiyama (Propositions 4.1 and 4.2). The present paper is organized as follows. First, we give various examples of BMO maps in §2. The main results of the present paper are given in §3. The following §4 is devoted to their proofs. Finally, in §5 we give a remark on BMO maps which are homeomorphisms between intervals.