Interpolation for Hardy Spaces: Marcinkiewicz decomposition, Complex Interpolation and Holomorphic Martingales

Abstract

The real and complex interpolation spaces for the classical Hardy spaces $H^1$ and $H^{\infty }$ 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 $L^1$ metric. Specifically for $f\in H^p$ the size of $$\inf \{\Vert f-f_1{\Vert }_1:f_1\in H^{\infty },\Vert f_1{\Vert }_{\infty }\le \lambda \}$$ needs to be determined for any $\lambda >0$. In the present paper we develop a new set of truncation formulae for obtaining the Marcinkiewicz decomposition of $(H^1,H^{\infty })$. We revisit the real and complex interpolation theory for Hardy spaces by examining our newly found formulae.