Fefferman multiplier theorem for Hardy martingales

A well-known theorem due to Fefferman provides a characterization of Fourier multipliers from $H^1(\mathbb{T})$ to ${\mathrm{\ell }}^1$, i.e. sequences ${\left({\lambda }_n\right)}_{n=0}^{\mathrm{\infty }}$ such that $$\sum _{n=0}^{\mathrm{\infty }}\mid {\lambda }_n\widehat{f}(n)\mid \lesssim \mathrm{\parallel }f{\mathrm{\parallel }}_{L^1(\mathbb{T})},$$ where $f(x)={\sum }_{n=0}^{\mathrm{\infty }}\widehat{f}(n)e^{inx}$. 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 ${\mathrm{\Delta }}_{n}f=f_n-f_{n-1}$ of the martingale wrt the standard filtration generated by $f$ satisfy $$(t\mapsto {\mathrm{\Delta }}_{n}f(x_1,\dots ,x_{n-1},t))\in H^1(\mathbb{T})\mathrm{.}$$ The key ingredient is a theorem due to P. F. X. M"uller stating that the classical Davis-Garsia decomposition $$\mathbb{E}{\left(\sum _{n=0}^{\mathrm{\infty }}{\mid {\mathrm{\Delta }}_{n}f\mid }^2\right)}^{\frac{1}{2}}\simeq \underset{f=g+h}{\inf }\mathbb{E}\sum _{n=0}^{\mathrm{\infty }}\mid {\mathrm{\Delta }}_{n}g\mid +\mathbb{E}{\left(\sum _{n=0}^{\mathrm{\infty }}\mathbb{E}({\mid {\mathrm{\Delta }}_{n}f\mid }^2\mid {\mathcal{F}}_{n-1})\right)}^{\frac{1}{2}}$$ may be done within the space of Hardy martingales.