Inégalité de Kato

Soit ${\displaystyle \Omega \subset \mathbb {R} ^{d}}$ un ensemble ouvert et borné et ${\displaystyle f\in L_{\operatorname {loc} }^{1}(\Omega )}$ , de sorte que ${\displaystyle \Delta f\in L_{\operatorname {loc} }^{1}(\Omega )}$. On a alors[2] - [3]

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

où[4]

${\displaystyle \operatorname {sgn} {\overline {f}}={\begin{cases}{\frac {\overline {f(x)}}{|f(x)|}}&{\text{si }}f\neq 0\\0&{\text{si }}f=0.\end{cases}}}$