Fractional embedding inequality with $L^{\infty}$ norm

Question

Here we consider the fractional Sobolev spaces and suppose $u$ is a vector function in $\mathbb R^2$. Is the following always true? $$\Vert Du \Vert_{L^{\infty}(\mathbb R^2)} \leq C\Vert Du \Vert^{1-\alpha}_{L^{q}(\mathbb R^2)}\Vert D^{\beta}u \Vert^{\alpha}_{L^{q}(\mathbb R^2)}$$

where we can calculate the relation between $\alpha$, $\beta$ and $q$ by scaling.