On low-dimensional uniform rectifiability in Heisenberg groups - Part 2

Let $1\le k\le n$. We prove that $k$-dimensional intrinsic Lipschitz graphs in the Heisenberg group ${\mathbb{H}}^n$ satisfy a geometric lemma $\mathrm{G}\mathrm{L}\mathrm{e}\mathrm{m}({\beta }_{2,{\mathcal{V}}_k},p)$ for horizontal $\beta$-numbers with an exponent $p=p(k)$. Previously, this result was known only in the case $k=1$; our proof recovers the sharp exponent $p=4$ in this setting. For $k>1$, we adapt an integral geometric approach originally developed by Orponen for Euclidean and parabolic Lipschitz functions. In addition, for $k=n$, we show how to deduce a geometric lemma directly from an isotropic Dorronsoro theorem in ${\mathbb{R}}^{2n}$ using a Morrey-type inequality. Building on the new geometric lemmas, we establish a necessary condition for $k$-regular sets in ${\mathbb{H}}^n$ to admit corona decompositions by intrinsic Lipschitz graphs. The condition is known to be sufficient by earlier work of the last two authors together with Pinamonti. It involves additional flatness coefficients besides ${\beta }_{2,{\mathcal{V}}_k}$. Along the way, we therefore extend the known stability results for geometric lemmas under the "big pieces’’ functor to a larger class of coefficients.