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]

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

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

where

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

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

• Sometimes the inequality is stated in the form

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

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

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

${\displaystyle \Delta |f|\geq \operatorname {Re} \left((\operatorname {sgn} {\overline {f}})\Delta f\right)\quad }${\displaystyle \Delta |f|\geq \operatorname {Re} \left((\operatorname {sgn} {\overline {f}})\Delta f\right)\quad } in ${\displaystyle \;{\mathcal {D}}'(\{f\neq 0\})}${\displaystyle \;{\mathcal {D}}'(\{f\neq 0\})}.^[6]

• 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年12月03日2.
• 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.

• Functional analysis
• Inequalities
• Differential operators

• Articles with short description
• Short description matches Wikidata