# Contents

## Idea

Kato’s inequality occurs in several areas of mathematics and physics. …

## Semigroup domination

Let $U_t = \exp(t H)$ be a positivity preserving semigroup on $L^p(M)$ for some fixed $p$. Let $V_t = \exp(t K)$ a strongly continuous semigroup on some vector valued $L^p(M,E)$, where $E$ is a Euclidian vector space with norm $\vert\cdot\vert$. For $a\in E$ set $(\sgn a\cdot v):=\langle a,v\rangle/\vert a\vert$. But you might as well take $E=\R$.

Theorem:

$\vert V_t a\vert \le U_t\vert a\vert \quad \forall t, a$

is equivalent to Kato’s inequality

$\langle \vert a\vert , H^\ast f\rangle \ge \langle (\sgn a)(Ka), f\rangle \quad \forall ..., f\ge0$

## Connection to positivity preserving semigroups

From Simon’s proof of the above theorem one easily sees that Kato’s inequality is necessary for any semigroup on $L^1$ to be positivity preserving.

Is it sufficient? – The other implication in Simon’s proof can easily be reduced to the criterion

which looks nice but ultimately begs the question in the language of resolvents. (For practical verification of the criterion you need to know the answer.) Using a crown jewel of semigroup tricks (from Dodziuk 19xy) the question can be begged more literally,

and this might perhaps turn out to guide the way to prove sufficiency: … tbc … Perhaps construct a positivity preserving semigroup dominated by $U_t$. Or perhaps just do some clever calculus, based perhaps on Egorov’s theorem

• B. Simon, Kato’s Inequality and the Comparison of Semigroups, J. Funct. Analysis 32 (1979), 92-101

• I. Shigekawa, $L^p$ contraction semigroups for vector valued functions

• Dodziuk

category: experiments