This is called a ‘smoothing property’ because $L^{\prime}$ quantitatively improves the regularity of $L$. It does not suggest $L^{\prime}$ is smooth, and indeed we expect the special Lagrangians which minimize $\mathcal{S}$ may have codimension two singularity. Since the volume is a priori bounded, the Hölder inequality shows that the $L^{p}$-smoothing property is stronger for bigger $p$, and in particular $L^{2}$-smoothing implies $L^{1}$-smoothing.