The local symmetry condition in the Heisenberg group

I propose an analogue in the first Heisenberg group $\mathbb{H}$ of David and Semmes’ local symmetry condition (LSC). For closed $3$-regular sets $E \subset $\mathbb{H}$$,Ishowthatthe(LSC)isimpliedbythe$L^{2}($\mathcal{H}$^{3}|_{E})$boundednessof$3$-dimensional singular integrals with horizontally antisymmetric kernels, and that the (LSC) implies the weak geometric lemma for vertical $\beta$-numbers.