Kato's inequality

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 ${\displaystyle \Omega \subset \mathbb {R} ^{d}}$ be a bounded and open set, and ${\displaystyle f\in L_{\operatorname {loc} }^{1}(\Omega )}$ such that ${\displaystyle \Delta f\in L_{\operatorname {loc} }^{1}(\Omega )}$. Then the following holds^[4][3]

${\displaystyle \Delta |f|\geq \operatorname {Re} \left((\operatorname {sgn} {\overline {f}})\Delta f\right)\quad }$ in ${\displaystyle \;{\mathcal {D}}'(\Omega )}$,

where

${\displaystyle \operatorname {sgn} {\overline {f}}={\begin{cases}{\frac {\overline {f(x)}}{|f(x)|}}&{\text{if }}f\neq 0\\0&{\text{if }}f=0.\end{cases}}}$^[5]

${\displaystyle L_{\operatorname {loc} }^{1}}$ 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

${\displaystyle \Delta f^{+}\geq \operatorname {Re} \left(1_{[f\geq 0]}\Delta f\right)\quad }$ in ${\displaystyle \;{\mathcal {D}}'(\Omega )}$

where ${\displaystyle f^{+}=\operatorname {max} (f,0)}$ and ${\displaystyle 1_{[f\geq 0]}}$ is the indicator function.

• If ${\displaystyle f}$ is continuous in ${\displaystyle \Omega }$ then

${\displaystyle \Delta |f|\geq \operatorname {Re} \left((\operatorname {sgn} {\overline {f}})\Delta f\right)\quad }$ in ${\displaystyle \;{\mathcal {D}}'(\{f\neq 0\})}$.^[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

1. Kato, Tosio (1972). "Schrödinger operators with singular potentials". Israel Journal of Mathematics. 13 (1–2): 135–148. doi:10.1007/BF02760233. S2CID 115546931.
2. 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.
3. 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.
4. 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.
5. Horiuchi, Toshio (2001). "Some remarks on Kato's inequality". Journal of Inequalities and Applications. 2001: 615789. doi:10.1155/S1025583401000030.
6. 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.