Interpolation for Hardy spaces: Marcinkiewicz decomposition, complex interpolation and holomorphic martingales
Volume 158 / 2019
Colloquium Mathematicum 158 (2019), 141-155
MSC: Primary 30H10, 30H05, 46B70; Secondary 60G46.
DOI: 10.4064/cm7460-10-2018
Published online: 29 July 2019
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 $$ {\rm{inf}} \{ \| f - f_1 \| _1 : f_1 \in H^\infty ,\, \|f_1\|_\infty \le \lambda \}$$ needs to be determined for any $\lambda \in \mathbb R_+. $ In the present paper we develop a new set of truncation formulae in order to obtain 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.
