$\alpha=\nu(p-1)$. Set $s={\psi}_{1}^{(-1)}(r)$, so that by definition of ${\psi}_{1}$ $$r^{-\alpha}{\psi}_{1}^{(-1)}(r)^{p-2}
    =
    s^{(1-\alpha)(p-2)}{\Lambda_p}(s^{-1})^{-\alpha}
    =
    [s^{-\frac{p}{{N}}}{\Lambda_p}(s^{-1})]^{-\alpha}
    \,.$$ By assumption the latter quantity is indeed nonincreasing in $s$ which however is a nondecreasing function of $r$.

\[l:bbl\] Under assumption we have that if $R$, $s>0$, $c\ge 1$ and $$\label{eq:bbl_m}
    R^{p}
    =
    cs
    {\psi}_{1}^{(-1)}(s^{-1})^{p-2}
    \,,$$ then $$\label{eq:bbl_mm}
    {\mu_{{\omega}}}(B_{{\lfloorR\rfloor}})
    \le
    \gamma(c)
    \psi_{1}^{(-1)}
    (s^{-1})^{-1}
    \,.$$

Let $\tau>0$ be such that $s^{-1}={\psi}_{1}(\tau)=\tau^{p-2}{\Lambda_p}(\tau^{-1})$. Then $$c^{-1}R^{p}
    =
    {\Lambda_p}(\tau^{-1})^{-1}
    \,.$$ On the other hand, on setting $v={\mu_{{\omega}}}(B_{{\lfloorR\rfloor}})$ and invoking we get $$c^{-1}R^{p}
    \ge
    c^{-1}{\mathcal{R}}(v)^{p}
    \ge
    c^{-1}
    \gamma_{0}
    {\Lambda_p}(v)^{-1}
    \ge
    {\Lambda_p}((\gamma_{0}c^{-1})^{\frac{{N}}{p}}v)^{-1}
    \,,$$ where we also used . Since ${\Lambda_p}$ is nonincreasing, the result follows.

Proofs of Proposition \[p:linf\_meas\] and Corollary \[co:linf\_int\] {#s:linf}
=====================================================================

By assumption, and by Remark \[r:lp\_scale\], ${u}\in L^{\infty}(0,T;\ell^{q}(V))$ for some $q> 1$; then for all $k>0$ the cut function ${{\ifthenelse{\equal{*}{*}}{({u}(t)-k)_{+}}{\left({u}(t)-k\right)_{+}}}}$ is finitely supported. For given $0<\sigma_{1}<\sigma_{2}<1/2$, $k>0$, $0<t<T$ define the decreasing sequences $$\begin{gathered}
    k_{i}
    =
    k[
    1-\sigma_{2}
    +
    2^{-i}
    (\sigma_{2}-\sigma_{1})
    ]
    \,,
    \qquad
    i=0\,,1\,,2\,,\dots
    \\
    t_{i}
    =
    \frac{t}{2}
    [
    1-\sigma_{2}
    +
    2^{-i}
    (\sigma_{2}-\sigma_{1})
    ]
    \,,
    \qquad
    i=0\,,1\,,2\,,\dots
  \end{gathered}$$ and let $f_{i}(x,\tau)={{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-k_{i})_{+}}{\left({u}(x,\tau)-k_{i}\right)_{+}}}}^{\nu}$, where $\nu=(p+q-2)/p$. Let also $$\begin{aligned}
    {m}_{i}(\tau)
    &=
      {\mu_{{\omega}}}(
      \{
      x\in V
      \mid
      {u}(x,\tau)>k_{i}
      \}
      )
      \,,
      \quad
      {M}_{i}
      =
      \sup_{t_{i}<\tau<t}
      {m}_{i}(\tau)
      \,,
    \\
    {{D}}_{i}(\tau)
    &=
      \sum_{x,y\in V}
      