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

TL;DR

This paper introduces new truncation formulas for the Marcinkiewicz decomposition in Hardy spaces, advancing the understanding of interpolation spaces and approximation problems within complex analysis.

Contribution

It develops novel truncation formulas for Hardy spaces, enabling refined analysis of interpolation and approximation in complex function spaces.

Findings

New truncation formulas for Hardy spaces

Enhanced understanding of interpolation spaces

Refined approximation techniques for Hardy spaces

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>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.