### A Marcinkiewicz type multiplier theorem for H¹ spaces on product domains

It is proved that if $m:{\mathbb{R}}^{d}\to \u2102$ satisfies a suitable integral condition of Marcinkiewicz type then m is a Fourier multiplier on the ${H}^{1}$ space on the product domain ${\mathbb{R}}^{{d}_{1}}\times ...\times {\mathbb{R}}^{{d}_{k}}$. This implies an estimate of the norm $N(m,{L}^{p}\left({\mathbb{R}}^{d}\right)$ of the multiplier transformation of m on ${L}^{p}\left({\mathbb{R}}^{d}\right)$ as p→1. Precisely we get $N(m,{L}^{p}\left({\mathbb{R}}^{d}\right))\lesssim {(p-1)}^{-k}$. This bound is the best possible in general.