Interpolation for Hardy spaces: Marcinkiewicz decomposition, complex interpolation and holomorphic martingales

The real and complex interpolation spaces for the classical Hardy spaces H1 and H∞ were determined in 1983 by P. W. Jones. Due to the analytic constraints the associated Marcinkiewicz decomposition gives rise to a delicate approximation problem for the L1 metric. Specifically for f∈Hp the size of inf{∥f−f1∥1:f1∈H∞,∥f1∥∞≤λ} needs to be determined for any λ∈R+. In the present paper we develop a new set of truncation formulae in order to obtain the Marcinkiewicz decomposition of (H1,H∞). We revisit the real and complex interpolation theory for Hardy spaces by examining our newly found formulae.