Weighted Carleman inequality for fractional gradient

We prove the weighted Carleman inequality for the fractional gradient
$$ \|e^{-t\langle a,{ \cdot }\rangle}|{ \cdot }|^{-\gamma}f\|_{q}\le C\|e^{-t\langle a,{ \cdot }\rangle}|{ \cdot }|^{\bar{\gamma}-\bar{\delta}}\nabla^{\alpha}f\|_{p}, f\in C_{0}^{\infty}(\mathbb{R}^{d}), t\ge 0. $$
For $\alpha=1$, it was proved by L. De Carli, D. Gorbachev, and S. Tikhonov (2020). An application of the Carleman inequality is given to prove the weak unique continuation property of a solution of the differential inequality with the potential $|\nabla^{\alpha}f|\le V|f|$ in a weighted Sobolev space.