A sufficient condition for Haar multipliers in Triebel-Lizorkin spaces

We consider Haar multiplier operators $T_m$ acting on Sobolev spaces, and more generally Triebel-Lizorkin spaces $F_{p,q}^s(\mathbb{R})$, for indices in which the Haar system is not unconditional. When $m$ depends only on the Haar frequency, we give a sufficient condition for the boundedness of $T_m$ in $F_{p,q}^s$, in terms of the variation norms $\mathrm{\parallel }m{\mathrm{\parallel }}_{V_u}$, which is optimal in $u$ (up to endpoints) when $p,q>1$.