Duhamel estimate with a common envelope

Statement

Let $U(t,s)$ be a propagator on $[0,L]$ , and suppose a positive function $P$ satisfies

\|U(t,s)\|\le C\frac{P(t)}{P(s)}\qquad(0\le s\le t\le L).

If $z'=A(t)z+g(t)$ , $z(0)=0$ , and $\|g(t)\|\le G P(t)$ , then

\|z(t)\|\le C G t P(t)\le C G L P(t).

Proof and hypotheses

The Duhamel formula gives $z(t)=\int_0^t U(t,s)g(s)\,ds$ , and the two envelope factors at $s$ cancel in the norm estimate. Nonzero initial data add at most $C P(t)\|z(0)\|/P(0)$ . A bound on values is proved here; parameter derivatives require differentiating the equation and controlling the resulting coefficient derivatives and source terms.