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.
The original inequality is for some degenerate elliptic operators. This article treats the special (but important) case for the Laplace operator.
Inequality for the Laplace operator
Let be a bounded and open set, and such that . Then the following holds
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 .