Bellman Function and the H 1 − BMO Duality

The emergence in the past decade of the Bellman function method as a powerful and versatile harmonic analysis technique has been characterized by rapid theoretical development on the one hand and somewhat ad hoc, if effective, approaches to some problems on the other. From the groundbreaking applications in [NTV1, NT, NTV2], which put the method on the map, to the concerted effort at tracing its origin to stochastic control and building a library of results in [NTV3, V] (see also multiple references therein; in addition, in [NTV3] an earlier result of Burkholder [B] was put in a Bellman-function framework), to recent explicit computation of actual Bellman functions (and not just their majorants) in [Va, M, VV, SV] – the technique has been established as one with many appearances and broad applicability. In this paper, we seek to reinforce this notion by using the Bellman function method in an unusual setting. Namely, we prove one, the more technically involved, direction of the famous Fefferman H1−BMO duality theorem ([F]). The proofs we present are Bellman-function-type proofs (see the discussion in [Sl]), whereas no extremal problem is posed and thus no Bellman function as such exists. Nonetheless, the main feature of any Bellman-function proof, an induction-by-scales argument, is central to our reasoning. (In Bellman-type arguments, the function on which the induction by scales is performed is commonly referred to as the Bellman function.) It is also worth noting that Bellman proofs often yield explicit (even sharp) constants in inequalities, one reason many well-known results have been reexamined recently with the use of the technique.