We prove that the $g$-function operator $g_\phi$, where $\phi(x)=h(|x|)\Omega(x)$ with $\Omega(x)=\Omega(x')\in H^1(S^{n-1})$ and $h(s)$ satisfing certain continuity hypothesis, is bounded on Triebel-Lizorkin space $F^{\alpha,q}_p(R^n)$ when $0 \lt \alpha \lt 1$ and $1 \lt p,q \lt \infty$. In particular, we get that the Marcinkiewicz integral operator $\mu_\Omega$ with $H^1$-kernel is bounded on $F^{\alpha,q}_p$
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