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 \[\sum_{n=0}^\infty \left|λ_n \widehat{f}(n)\right|\lesssim \|f\|_{L^1(\mathbb{T})},\] where $f(x)={\sum }_{n=0}^{\infty }\hat{f}(n)e^{inx}$. 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 ${\Delta }_{n}f=f_n-f_{n-1}$ of the martingale wrt the standard filtration generated by $f$ satisfy \[\left(t\mapsto Δ_n f\left(x_1,\ldots,x_{n-1},t\right)\right)\in H^1(\mathbb{T}). \] The key ingredient is a theorem due to P. F. X. Müller stating that the classical Davis-Garsia decomposition \[\mathbb{E} \left(\sum_{n=0}^\infty \left|Δ_n f\right|^2\right)^\frac{1}{2}\simeq \inf_{f=g+h} \mathbb{E}\sum_{n=0}^\infty \left|Δ_n g\right|+ \mathbb{E}\left(\sum_{n=0}^\infty \mathbb{E}\left(\left|Δ_n f\right|^2\mid \mathcal{F}_{n-1}\right)\right)^\frac{1}{2}\] may be done within the space of Hardy martingales.