Holomorphic functions, measures and BMO

Abstract

We extend the above theorem to the general case of finite strictly positive continuous measures on T, under the supplementary restriction that f (zo)r ). In the particular case where /l is the Lebesgue measure, Theorem 1 implies that the hypothesis '3r(z0)r ' ' is not needed. However, this restriction is not superfluous in the general case; see w 4, prop. 19 for a relevant counterexample. The above extension is purely topological in nature. We prove that for any complex continuous func t ionfon T and any complex number w Cf(T), the following are equivalent: a) For every finite strictly positive continuous measure # on T, there is an interval I c T such that w= 1//z(I) fxfdl~. b) f has non-zero winding number with respect to w. This equivalence enables us to determine the range of the BMO norm of q~o U, where q~ is any given continuous unimodular function on T and U varies in the set of all homeomorphisms of T onto itself. In the case where r has non-zero winding number with respect to 0, we show that (0o U has BMO norm equal to 1 for all U. If q9 has zero winding number with respect to 0, then the BMO norm of q)oU can be made arbitrarily close to zero and does not exceed