Weighted gradient inequalities and unique continuation problems

Abstract

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