Let $N\in\mathbb N$, and let $M\subset\mathbb C^N$ be a $C^\infty$ embedded generic CR submanifold. Let $\mathcal O\subset M$ be a connected CR orbit which is open in $M$, equipped with the induced CR structure and the restricted [Hausdorff measure](/page/Hausdorff%20Measure) $\mathcal H^d|_{\mathcal O}$, where $d:=\dim_{\mathbb R}M$. Assume the following Baouendi-Treves local propagation principle holds on $\mathcal O$: there is a family $\mathscr P$ of connected generic CR coordinate patches $P\Subset\mathcal O$ covering $\mathcal O$ such that, whenever $P\in\mathscr P$, $g\in L^2_{\mathrm{loc}}(P,\mathcal H^d;\mathbb C)$ is a distributional CR function, and $g=0$ $\mathcal H^d$-a.e. on a non-empty open subset $A\subset P$, then $g=0$ $\mathcal H^d$-a.e. on every relatively compact connected open subpatch $P_0\Subset P$ contained in the local CR orbit of $A$ in $P$. Assume also that, for every $P\in\mathscr P$ and every non-empty open subset $A\subset P$, each point of the local CR orbit of $A$ in $P$ has a relatively compact connected open neighbourhood $P_0\Subset P$ contained in that local CR orbit. Assume also that every piecewise $C^1$ curve in $\mathcal O$ whose tangent vector lies in the real CR distribution whenever the derivative exists admits a finite subdivision whose subarcs lie in patches from $\mathscr P$, with the two endpoints of each subarc lying in the same local CR orbit of the corresponding patch. Let

\begin{align*} f\in L^2_{\mathrm{loc}}(\mathcal O,\mathcal H^d;\mathbb C) \end{align*}

be a distributional CR function on $\mathcal O$. If there exists a non-empty [open set](/page/Open%20Set) $U\subset\mathcal O$ such that $f=0$ $\mathcal H^d$-a.e. on $U$, then, for every compact set $K\subset\mathcal O$, one has $f=0$ $\mathcal H^d$-a.e. on $K$. Equivalently, $f=0$ $\mathcal H^d$-a.e. on $\mathcal O$.