Dimension Independent Atomic Decomposition for Dyadic Martingale $$ \pmb {{\mathbb {H}}}^{1}$$ H 1

We introduce atoms for dyadic atomic $${\mathbb {H}}^1$$ for which the equivalence between the atomic and maximal function definitions is dimension independent. We give sharp, up to $$\log (d)$$ factor, estimates for the $${{\mathbb {H}}^1}\rightarrow L^1$$ norm of the special maximal function.