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.

中文翻译：

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加权梯度不等式和唯一的连续问题

我们将Pitt不等式用于Fourier变换以证明以下加权梯度不等式

$$ \ begin {aligned} \ Vert e ^ {-\ tau \ ell（\ cdot）} u ^ {\ frac {1} {q}} f \ Vert _q \ le c_ \ tau \ Vert e ^ {-\ tau \ ell（\ cdot）} v ^ {\ frac {1} {p}} \，\ nabla f \ Vert _p，\ quad f \ in C ^ \ infty _0（{\ mathbb {R}} ^ n） 。\ end {aligned} $$

该不等式是Carleman型估计，对于一阶微分方程和系统的解产生唯一的连续结果。