Where can i find proof of this Lemma? $S(t)$ is the heat semigroup. If $p \geq 1$, then $[S(t)u_0]^p \leq S(t)u^p_0$

Question

The Lemma below is valid for the heat semigroup

Lemma. Assume that $u_0 \in C_0(Ω)$, $u_0 \geq 0$. If $p \geq 1$, then $[S(t)u_0]^p \leq S(t)u^p_0$. If $0 < p < 1$, then $[S(t)u_0]^p \geq S(t)u^p_0$.

does anyone know where i can find a proof of this? Also, is there a similar version for Schrödinger semigroups? If so, where can I find it?