In this paper we consider the n-dimensional Euclidean space equipped with anisotropic dilations and let Q be the homogeneous dimension. We introduce the anisotropic fractional differential operator \(\mathscr {D} ^s,s>0,\) and prove the following anisotropic fractional Leibniz rule: \(\begin{aligned} {\left\| \mathscr {D} ^{s}(f g)\right\| _{L^{r}} \lesssim \left\| \mathscr {D} ^{s} f\right\| _{L^{p}}\Vert g\Vert _{L^{q}}+\Vert f\Vert _{L^{p}}\left\| \mathscr {D} ^{s} g\right\| _{L^{q}},} \end{aligned}\) where \(r \in \left[ \frac{1}{2}, \infty \right] \) , \(p, q \in [1, \infty ]\) satisfy \(\frac{1}{r}=\frac{1}{p}+\frac{1}{q}\) , \( s>\max (\frac{Q}{r}-Q, 0)\) or \(s \in 2 \mathbb {N}\) , and \(f,g\in \mathcal {S}(\mathbb {R}^n)\) . We also show that the above range of s is sharp.
