In functional analysis, a subfield of mathematics, Kato's inequality is a distributional inequality for the Laplace operator or certain elliptic operators. It was proven in 1972 by the Japanese mathematician Tosio Kato.[1]
The original inequality is for some degenerate elliptic operators.[2] This article treats the special (but important) case for the Laplace operator.[3]
Inequality for the Laplace operator
Let be a bounded and open set, and such that . Then the following holds[4][3]
- in ,
where
is the space of locally integrable functions – i.e., functions that are integrable on every compact subset of their domains of definition.
Remarks
- Sometimes the inequality is stated in the form
- in
- where and is the indicator function.
- If is continuous in then
- in .[6]
Literature
- Brezis, Haı̈m; Ponce, Augusto (2004). "Kato's inequality when Δu is a measure". Comptes Rendus Mathematique. 338 (8): 599–604. doi:10.1016/j.crma.2003.12.032.
- Arendt, Wolfgang; ter Elst, Antonious F.M. (2019). "Kato's Inequality". Analysis and Operator Theory. Springer Optimization and Its Applications. Springer Optimization and Its Applications. Vol. 146. Cham: Springer. pp. 47–60. doi:10.1007/978-3-030-12661-2_3. ISBN 978-3-030-12660-5. S2CID 191796248.
References
- ^ Kato, Tosio (1972). "Schrödinger operators with singular potentials". Israel Journal of Mathematics. 13 (1–2): 135–148. doi:10.1007/BF02760233. S2CID 115546931.
- ^ Devinatz, Allen (1979). "On an Inequality of Tosio Kato for Degenerate-Elliptic Operators". Journal of Functional Analysis. 32 (3): 312–335. doi:10.1016/0022-1236(79)90043-0.
- ^ a b Brezis, Haı̈m; Ponce, Augusto (2004). "Kato's inequality when Δu is a measure". Comptes Rendus Mathematique. 338 (8): 599–604. doi:10.1016/j.crma.2003.12.032.
- ^ Arendt, Wolfgang; ter Elst, Antonious F.M. (2019). "Kato's Inequality". Analysis and Operator Theory. Springer Optimization and Its Applications. Springer Optimization and Its Applications. Vol. 146. Cham: Springer. pp. 47–60. doi:10.1007/978-3-030-12661-2_3. ISBN 978-3-030-12660-5. S2CID 191796248.
- ^ Horiuchi, Toshio (2001). "Some remarks on Kato's inequality". Journal of Inequalities and Applications. 2001: 615789. doi:10.1155/S1025583401000030.
- ^ Dávila, Juan; Ponce, Augusto (2003). "Variants of Kato's inequality and removable singularities". Journal d'Analyse Mathématique. 91: 143–178. doi:10.1007/BF02788785. S2CID 55929478.