Fefferman multiplier theorem for Hardy martingales

Abstract

A well-known theorem due to Fefferman provides a characterization of Fourier multipliers from $H^1(\mathbb{T})$ to $\ell^1$, i.e. sequences $\left(\lambda_n\right)_{n=0}^\infty$ such that \[\sum_{n=0}^\infty \left|\lambda_n \widehat{f}(n)\right|\lesssim \|f\|_{L^1(\mathbb{T})},\] where $f(x)=\sum_{n=0}^\infty \widehat{f}(n)e^{inx}$. We extend it to the space $H^1\left(\mathbb{T}^\mathbb{N}\right)$ of Hardy martingales, i.e. the subspace of $L^1$ on the countable product $\mathbb{T}^\mathbb{N}$ consisting of all $f$ such that the differences $\Delta_nf=f_{n}-f_{n-1}$ of the martingale wrt the standard filtration generated by $f$ satisfy \[\left(t\mapsto \Delta_n f\left(x_1,\ldots,x_{n-1},t\right)\right)\in H^1(\mathbb{T}). \] The key ingredient is a theorem due to P. F. X. M\"uller stating that the classical Davis-Garsia decomposition \[\mathbb{E} \left(\sum_{n=0}^\infty \left|\Delta_n f\right|^2\right)^\frac{1}{2}\simeq \inf_{f=g+h} \mathbb{E}\sum_{n=0}^\infty \left|\Delta_n g\right|+ \mathbb{E}\left(\sum_{n=0}^\infty \mathbb{E}\left(\left|\Delta_n f\right|^2\mid \mathcal{F}_{n-1}\right)\right)^\frac{1}{2}\] may be done within the space of Hardy martingales.