Kato's inequality

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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]

• 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. hdl:2078.1/70476.
• 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.