Geodesic
Weighted Carleman inequality for fractional gradient
Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 152-156.

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We prove the weighted Carleman inequality for the fractional gradient $$ \|e^{-t\langle a,{ \cdot }\rangle}|{ \cdot }|^{-\gamma}f\|_{q}\le C\|e^{-t\langle a,{ \cdot }\rangle}|{ \cdot }|^{\bar{\gamma}-\bar{\delta}}\nabla^{\alpha}f\|_{p}, f\in C_{0}^{\infty}(\mathbb{R}^{d}), t\ge 0. $$ For $\alpha=1$, it was proved by L. De Carli, D. Gorbachev, and S. Tikhonov (2020). An application of the Carleman inequality is given to prove the weak unique continuation property of a solution of the differential inequality with the potential $|\nabla^{\alpha}f|\le V|f|$ in a weighted Sobolev space.
Keywords: Carleman's inequality, fractional gradient, Fourier transform, Pitt's inequality, differential inequality. 
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     author = {D. V. Gorbachev},
     title = {Weighted {Carleman} inequality for fractional gradient},
     journal = {\v{C}eby\v{s}evskij sbornik},
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     publisher = {mathdoc},
     volume = {23},
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     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a12/}
}
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D. V. Gorbachev. Weighted Carleman inequality for fractional gradient. Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 152-156. http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a12/

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