Weighted gradient inequalities and unique continuation problems

Abstract

We use Pitt inequalities for the Fourier transform to prove the following weighted gradient inequality $$\begin{aligned} \Vert e^{-\tau \ell (\cdot )} u^{\frac{1}{q}} f\Vert _q\le c_\tau \Vert e^{-\tau \ell (\cdot )} v^{\frac{1}{p}}\, \nabla f\Vert _p, \quad f\in C^\infty _0({\mathbb {R}}^n). \end{aligned}$$This inequality is a Carleman-type estimate that yields unique continuation results for solutions of first order differential equations and systems.