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Let $g:\mathbb{R}^3\to\mathbb{R^2}$ be a continuous function. Suppose that there exists $\Omega$ a neighborhood of $0$ where $Xg, Yg \in L^\infty(\Omega)$, with $$X=\partial_x-\frac{y}{2}\partial_z\qquad Y=\partial_y+\frac{x}{2}\partial_z$$ and $Z=[X,Y]=\partial_z$.

Now, let $J=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ and set $D_g=(Xg\ \vert Yg)$. The distribution $$[X((Yg)^tJg)-Y((Xg)^tJg)]-2\det D_g=T_g$$ is well-defined and one can check that, if $g\in\mathcal{C}^1$, then $T_g$ can be represented by integration against $(Zg)^tJg$.

My question is the following: is it true that the functions $$G_\lambda(x,y,z)=\frac{1}{\lambda}(g(x,y,z+\lambda)-g(x,y,z))^tJg(x,y,z)$$ converge (in the sense of distributions) to $T_g$ as $\lambda$ goes to $0$?


I tried to approximate $g$ by smooth functions $g_\epsilon$, but then I find myself dealing with a double limit: $G_{\epsilon,\lambda}\to T_g$ as $(\epsilon,\lambda)\to(0,0)$ and I don't know how to do it.

Obviously, the main problem is that the limit does not involve the incremental ratio alone, but rather a function of it, and there is no distributional meaning of $(Zg)^tJg$ for $g$ merely continuous (to my best knowledge, at least).

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