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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 analyticconstraints the associated Marcinkiewicz decomposition gives rise to a delicateapproximation problem for the $L^ 1$ metric. Specifically for $ f \in H^p$ thesize of $$ {\rm{inf}} \{ \| f - f_1 \| _1 \,:\, f_1 \in H^\infty ,\,\|f_1\|_\infty \le \lambda \}$$ needs to be determined for any $ \lambda>0 $.In the present paper we develop a new set of truncation formulae for obtainingthe Marcinkiewicz decomposition of $(H^1, H^\infty) $. We revisit the real andcomplex interpolation theory for Hardy spaces by examining our newly foundformulae.

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