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.
Categories: math.PR math.FA