Weighted Carleman inequality for fractional gradient
https://doi.org/10.22405/2226-8383-2022-23-4-152-156
Abstract
We prove the weighted Carleman inequality for the fractional gradient
‖𝑒−𝑡⟨𝑎, · ⟩| · |−𝛾𝑓‖𝑞 <= 𝐶‖𝑒−𝑡⟨𝑎, · ⟩| · |¯𝛾−¯𝛿∇𝛼𝑓‖𝑝, 𝑓 ∈ 𝐶∞
0 (R𝑑), 𝑡 > 0.
For 𝛼 = 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 |∇𝛼𝑓| <= 𝑉 |𝑓| in a weighted Sobolev space.
Keywords
Carleman’s inequality,
fractional gradient,
Fourier transform,
Pitt’s inequality,
differential inequality.
Supporting agencies: This Research was performed by a grant of Russian Science Foundation (project 18-11-00199), https://rscf. ru/project/18-11-00199/.
About the Author
Dmitriy Victorovich Gorbachev
Tula State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
References
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3. De Carli, L., Gorbachev, D. & Tikhonov, S. 2020. “Weighted gradient inequalities and unique
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Review
For citations:
Gorbachev D.V. Weighted Carleman inequality for fractional gradient. Chebyshevskii Sbornik. 2022;23(4):152-156. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-4-152-156
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