Conditional Square Functions, the Sine-Cosine Decomposition for Hardy Martingales and Dyadic Perturbation

Abstract

We prove that the $\cal P$ norm estimate between a Hardy martingale and its cosine part are stable under dyadic perturbations, and show how dyadic stability of the $\cal P$ norm estimate is used in the proof that $L^1$ embeds into $L^1/H^1$.