Weighted gradient inequalities and unique continuation problems

We use Pitt inequalities for the Fourier transform to prove the following weighted gradient inequality $$\mathrm{\parallel }{e}^{-\tau \mathrm{\ell }(\cdot )}{u}^{\frac{1}{q}}f{\mathrm{\parallel }}_q\le c_{\tau }\mathrm{\parallel }{e}^{-\tau \mathrm{\ell }(\cdot )}{v}^{\frac{1}{p}}\mathrm{\nabla }f{\mathrm{\parallel }}_p,f\in C_0^{\mathrm{\infty }}(R^n)\mathrm{.}$$ This inequality is a Carleman-type estimate that yields unique continuation results for solutions of first order differential equations and systems.