Weak type (1,1) in a Refinement of the Marcinkiewicz Theorem

Let $m$ be a bounded function on $\mathbb{R}_+$ whose $p$-variations on the intervals $[2^k,2^{k+1}]$, $k\in\mathbb{Z}$, are uniformly bounded for some $p<2$. Then the operator $T$, $\widehat{Tf}=m\hat f$, is of weak type $(1,1)$ on the space $H^1(\mathbb{R})$.