A Marcinkiewicz type multiplier theorem for H¹ spaces on product domains

It is proved that if ${\displaystyle m:{ℝ}^d\to ℂ}$ satisfies a suitable integral condition of Marcinkiewicz type then m is a Fourier multiplier on the ${\displaystyle H^1}$ space on the product domain ${\displaystyle {ℝ}^{d_1}\times ...\times {ℝ}^{d_k}}$. This implies an estimate of the norm ${\displaystyle N(m,L^p\left({ℝ}^d\right)}$ of the multiplier transformation of m on ${\displaystyle L^p\left({ℝ}^d\right)}$ as p→1. Precisely we get ${\displaystyle N(m,L^p\left({ℝ}^d\right))\lesssim {(p-1)}^{-k}}$. This bound is the best possible in general.