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,2r))⩽Cμ(B(x,r)). We generalize Wolffʹs 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). Wolffʹs 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 Wolffʹs theorem must be dropped if T is replaced by S2n−1 with n⩾2. Move precisely, if n⩾2, then there is a g∈L∞(S2n−1,σ), where σ is the standard spherical measure on S2n−1, such that if q∈H∞(S2n−1) and if q is not the constant function 0, then gq does not have vanishing mean oscillation on S2n−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 S2n−1 when n⩾2. We conclude the paper with an index theorem for Toeplitz operators on S2n−1.
