  We use Pitt inequalities for the Fourier transform to prove the following
weighted gradient inequality $$ \|e^{-\tau\ell(\cdot)} u^{\frac 1q} f\|_q\leq
c_\tau\| e^{-\tau\ell(\cdot)} v^{\frac 1p}\, \nabla f\|_p, \quad f\in
C^\infty_0( R^n). $$ This inequality is a Carleman-type estimate that yields
unique continuation results for solutions of first order differential equations
and systems.
