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Weighted gradient inequalities and unique continuation problems

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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.

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Authors and Affiliations

  1. Department of Mathematics, Florida International University, Miami, FL, 33199, USA

    Laura De Carli

  2. Department of Applied Mathematics and Computer Science, Tula State University, Tula, Russia, 300012

    Dmitry Gorbachev

  3. ICREA, Centre de Recerca Matemàtica, UAB, Campus de Bellaterra, Edifici C, 08193, Bellaterra, Barcelona, Spain

    Sergey Tikhonov

Authors
  1. Laura De Carli
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  2. Dmitry Gorbachev
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  3. Sergey Tikhonov
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Correspondence to Laura De Carli.

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Communicated by L. Ambrosio.

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D. G. was supported by the Russian Science Foundation under Grant 18-11-00199. S. T. was partially supported by MTM 2017-87409-P, 2017 SGR 358, and by the CERCA Programme of the Generalitat de Catalunya.

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De Carli, L., Gorbachev, D. & Tikhonov, S. Weighted gradient inequalities and unique continuation problems. Calc. Var. 59, 89 (2020). https://doi.org/10.1007/s00526-020-1716-8

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