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Weighted gradient inequalities and unique continuation problems. (English) Zbl 1439.42009

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

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B37 Harmonic analysis and PDEs
35B60 Continuation and prolongation of solutions to PDEs
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