$L^{\infty}$-truncation of closed differential forms

In this paper, we prove that for each closed differential form $u \in L^1(\mathbb{R}^N,(\mathbb{R}^N)^{\ast} \wedge ... \wedge (\mathbb{R}^N)^{\ast})$, which is almost in $L^{\infty}$ in the sense that $$\int_{\{y \in \mathbb{R}^N \colon \vert u(y) \vert \geq L \}} \vert u(y) \vert dy < \varepsilon$$ for some $L>0$ and a small $\varepsilon >0$, we may find a closed differential form $v$, such that $\Vert u - v \Vert_{L^1}$ is again small, and $v$ is, in addition, in $L^{\infty}$ with a bound on its $L^{\infty}$ norm depending only on $N$ and $L$. In particular, the set $\{ v \neq u\}$ has measure at most $C \varepsilon$. We then look at applications of this theorem. We are able to prove that the $\mathcal{A}$-$p$-quasiconvex hull of a set does not depend on $p$. Furthermore, we can prove a classification theorem for $\mathcal{A}$-$\infty$-Young measures.