($L^{p}$-smoothing property) There exists $p\geq 1$ and a uniform constant $C$, such that for any $L\in\mathcal{L}$, we can find $L^{\prime}\in\mathcal{L}$ with $\mathcal{S}(L^{\prime})\leq\mathcal{S}(L)$, and $\int_{L^{\prime}}|\vec{H}|^{p}\leq C$.