Bounded functions of vanishing mean oscillation on compact metric spaces

Abstract

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,2 r))⩽ 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 2 n−1 with n⩾2. Move precisely, if n⩾2, then there is a g∈ L ∞( S 2 n−1 , σ), where σ is the standard spherical measure on S 2 n−1 , such that if q∈ H ∞( S 2 n−1 ) and if q is not the constant function 0, then gq does not have vanishing mean oscillation on S 2 n−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 2 n−1 when n⩾2. We conclude the paper with an index theorem for Toeplitz operators on S 2 n−1 .