rooro

The following lemma is a variable exponent modification of~\cite[Chapter 4.7.1, Lemma 2(iii)]{EvaG92}.
\begin{lemma} \label{subadd_apulemma}
Let $u_i\in\W(\Rn)$ and define
    \begin{align*}
        & u(x) := \sup\limits_{i} u_i(x) , \\
        & g(x) := \sup\limits_{i} |\nabla u_i(x)|.
    \end{align*}
If $u,g\in\Lp(\Rn)$, then $u\in\W(\Rn)$ and $|\nabla u|\leq g$ almost everywhere.
\end{lemma}

\begin{proof}
The variable exponent H{\" o}lder inequality \cite[Lemma 3.2.20]{DieHHR11} implies that $u$ and
$g$ are locally integrable functions.
Let us now recall the proof of \cite[Chapter 4.7.1, Lemma 2(iii)]{EvaG92}. The main tools are the Riesz representation of a functional and the Radon-Nikodym theorem. The methods of applying the Riesz representation and the Radon-Nikodym theorem remain valid also in our case. Even though we work from slightly different function spaces.
Finally, we have that $u$ has the weak gradient $\nabla u$ dominated by $g$. This also completes the statement about $u$ belonging to the desired Sobolev space.
\end{proof}

\begin{huomautus}
See also \cite[Lemma 1.28]{BjoB_pp09} for a very short proof of a similar result in the metric measure space setting. In the proof, the supremum of the upper gradients corresponds to the function $g$ in our lemma.
\end{huomautus}