and $\frac{\max(|U'|)}{\min(|U'|)}<C$ by condition \[H3\]. We may hence employ a change of variables $z=U_{\star}(y)$ and define a Fourier multiplier with respect to $z$ as in [@Zill5], where the case of a bilipschitz profile was analyzed in detail.

With this preparation, we can now prove our first main result.

In view of \[H3\], $U$ is Bilipschitz on each interval $I_j$. We thus introduce $A_j(t)$ as given in Lemma \[lem:Bilipschitz\] and define $$\begin{aligned}
    A(t)= \sum_j \chi_j A_j(t) \chi_j.
  \end{aligned}$$ Then by Proposition \[prop:reduction\] it holds that $$\begin{aligned}
    \ddt \l \omega, A(t) \omega\r \leq \Re\l \omega, \dot A (t) \omega \r +C\eps_0\sum_{j\in \J} \int_{I_j} |U'(y)| \left[|\nabla_k  \psi_j(y)|^2 +k^2|\nabla_k \psi_{j,A}(y) |^2\right]\dd y.
  \end{aligned}$$ Exploiting , it follows that $$\begin{aligned}
     \ddt \l \omega, A(t) \omega \r \leq (C\eps_0-1)\sum_{j\in \J}\int_{I_j} |U'(y)| \left[|\nabla_k  \psi_j(y)|^2 +k^2|\nabla_k \psi_{j,A}(y) |^2\right]\dd y. \end{aligned}$$ Therefore, given and \[P\], we infer that $$\begin{aligned}
     \ddt \l \omega, A(t) \omega \r + \frac12\sum_{j\in \J}\int_{I_j} |U'(y)| \left[|\nabla_k  \psi_j(y)|^2 +k^2|\nabla_k \psi_{j,A}(y) |^2\right]\dd y\leq0.
     \end{aligned}$$ From this, we deduce that $t\mapsto \l \omega(t), A(t) \omega(t) \r$ is non-increasing. Furthermore, since $0 \leq \l \omega(t), A(t)  \omega(t) \r$ for all $t\geq 0$, this also implies that $$\begin{aligned}
    \int_{0}^T\left[\sum_{j\in \J}\int_{I_j} |U'(y)| \left[|\nabla_k  \psi_j(y)|^2 +k^2|\nabla_k \psi_{j,A}(y) |^2\right]\dd y \right]\dd t \leq 