Bounded functions of vanishing mean oscillation on compact metric spaces

A well-known theorem of Wolff (Duke Math. J 49 (1982) 321) asserts that for every f ∈ L^∞ on the unit circle T, there is a non-trivial q ∈ QA = VMO∩H^∞ such that fq∈QC. In this paper we consider the situation where T is replaced by a compact metric space (X, d) equipped with a measure μ satisfying the condition μ(B(x, 2r)) ≤ Cμ(B(x, r)). We generalize Wolff's theorem to the extent that every function in L^∞ (X, μ) can be multiplied into VMO(X, d, μ) in a non-trivial way by a function in VMO(X, d, μ ∩L^∞ (X, μ). Wolff's proof relies on the fact that T has a dyadic decomposition. But since this is not available for (X, d) in general, our approach is completely different. Furthermore, we show that the analyticity requirement for the function q in Wolff's theorem must be dropped if T is replaced by S^2n-1 with n≥2. Move precisely, if n≥2, then there is a g∈H^∞ (S^2n-1, σ), where σ is the standard spherical measure on S^2n-1, such that if q∈H^∞ (S^2n-1) and if q is not the constant function 0, then gq does not have vanishing mean oscillation on S^2n-1. The particular g that we construct also serves to show that a famous factorization theorem of Axler (Ann. of Math. 106 (1977) 567) for L^∞-functions on the unit circle T cannot be generalized to S^2n-1 when n≥2. We conclude the paper with an index theorem for Toeplitz operators on S^2n-1.