Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups (Q6621687)

The authors study regularity properties for the intrinsic graphs of functions in Carnot groups.\N\NLet \(\mathbb{G}\) be a Carnot group, equipped with a homogeneous norm \(\|\cdot\|\). Fix a \textit{splitting} \(\mathbb{G} = \mathbb{W} \cdot \mathbb{L}\), where \(\mathbb{W}, \mathbb{L} \subseteq \mathbb{G}\) are two subgroups which are homogeneous (i.e., closed under dilation), with \(\mathbb{W} \cap \mathbb{L} = \{e\}\); let \(\pi_{\mathbb{W}}, \pi_{\mathbb{L}}\) be the corresponding projections. We further suppose that \(\mathbb{L}\) is horizontal, i.e., a subset of the first layer of \(\mathbb{G}\). We consider a function \(\varphi : U \subset \mathbb{W} \to \mathbb{L}\), where \(U \subset \mathbb{W}\) is relatively open. The intrinsic graph of \(\varphi\) is taken to be the set \(\operatorname{graph}(\varphi) = \{ p \cdot \varphi(p) : p \in U \} \subseteq \mathbb{G}\).\N\N(This extends the case of a function \(\varphi : \mathbb{R}^n \to \mathbb{R}^m\), for which we could take \(\mathbb{G} = \mathbb{R}^{n+m}\), \(\mathbb{W} = \mathbb{R}^n \oplus 0\), \(\mathbb{L} = 0 \oplus \mathbb{R}^m\). A simple non-abelian example is \(\mathbb{G} = \mathbb{H}^1\) the three-dimensional Heisenberg group, for which we could have, in the usual coordinates \((x,y,z)\), the splitting \(\mathbb{W} = \{(x,0,z) : x,z \in \mathbb{R} \}\) and \(\mathbb{L} = \{(0,y,0) : y \in \mathbb{R}\}\).)\N\NThe main result of the paper (Theorem 1.6) is the equivalence of the following four conditions. For brevity, here I will only summarize them informally, and refer the reader to the paper for the precise technical statements and associated definitions.\N\begin{itemize}\N\item[(a)] The graph of \(\varphi\) is a co-horizontal \(C^1_H\)-surface with homogeneous tangent spaces complemented by \(\mathbb{L}\).\N\NThis means that it is locally the zero set of an \(\mathbb{R}^k\)-valued function \(f\) with continuous and surjective Pansu differential \(\mathrm{d_P}f\). The homogeneous tangent space at \(p \in \operatorname{graph}(\varphi)\) is defined as the kernel of the group homomorphism \(\mathrm{d_p}f(p) : \mathbb{G} \to \mathbb{R}^k\), which is thus a homogeneous subgroup of \(\mathbb{G}\); it is asserted that this subgroup is complemented by \(\mathbb{L}\).\N\N\item[(b)] \(\varphi\) is uniformly intrinsically differentiable. This notion of differentiability was introduced in [\textit{L. Ambrosio} et al., J. Geom. Anal. 16, No. 2, 187--232 (2006; Zbl 1085.49045)]; unlike Pansu differentiability, it involves the splitting \(\mathbb{G} = \mathbb{W} \cdot \mathbb{L}\) in an explicit way, so that it depends not only on \(\mathbb{W}, \mathbb{L}\) as groups but also on their embedding into \(\mathbb{G}\).\N\N\item[(c)] \(\varphi\) is vertically broad* hölder (see below) and is locally uniformly approximated by a family of \(C^1\) functions \(\varphi_\epsilon\) whose derivatives along certain horizontal projected left-invariant vector fields also converge locally uniformly.\N\NThe vector fields used in this statement are constructed by starting with a left-invariant vector field \(W \in \operatorname{Lie}(\mathbb{W}) \subset \operatorname{Lie}(\mathbb{G})\), restricting it to \(\operatorname{graph}(\varphi)\), and projecting it to \(\mathbb{W}\) via \(\pi_{\mathbb{W}}\) (which, we note again, depends on the splitting \(\mathbb{G} = \mathbb{W} \cdot \mathbb{L}\)). The vertically broad* hölder regularity condition relates to the Hölder regularity of \(\varphi\) along the integral curves of such projected vector fields.\N\N\item[(d)] \(\varphi\) is vertically broad* hölder in the same sense as above, and is differentiable in the broad* sense. This latter property is defined in terms of an absolute continuity condition along the integral curves of projected vector fields as above.\N\end{itemize}\N\NA version of the equivalence of (a) and (b) was previously shown in [\textit{D. Di Donato}, Potential Anal. 54, No. 1, 1--39 (2021; Zbl 1456.35206)], and for Heisenberg groups in [\textit{L. Ambrosio} et al., J. Geom. Anal. 16, No. 2, 187--232 (2006; Zbl 1085.49045)] (under stronger conditions) and in [\textit{G. Arena} and \textit{R. Serapioni}, Calc. Var. Partial Differ. Equ. 35, No. 4, 517--536 (2009; Zbl 1225.53031)]. The equivalence of (a) and (d) appeared in [Kozhevnikov, Artem, Propriétés métriques des ensembles de niveau des applications différentiables sur les groupes de Carnot, 2015], and the equivalence of (b) and (d), in the setting of Heisenberg groups, appeared in [\textit{F. Bigolin} and \textit{F. Serra Cassano}, Adv. Calc. Var. 3, No. 1, 69--97 (2010; Zbl 1188.53027)].\N\NIn the special case where \(\mathbb{G}\) has step 2 and \(\mathbb{L}\) is one-dimensional, the above result is strengthened (Theorem 1.7) by showing that each of items (c) and (d) still imply the others even without the condition of vertical broad* Hölder regularity. However, the authors mention that an example given in [Kozhevnikov, op cit], for the Engel group, shows that such a strengthening cannot hold in general.\N\NAs a side result, the authors provide an area formula for UID functions \(\varphi\) (Proposition 1.8), expressing the \(\mathbb{G}\)-perimeter measure of its subgraph in terms of an integral involving the intrinsic gradient of \(\varphi\).