The coarea formula states that any locally Lipschitz function (e.g. a $C^1$-function) $F:\mathbb{R}^N\to\mathbb{R}^n$ with $N\geq n$ satisfies $$\int_A JF(x) \mathrm{d}\mathcal{H}^N = \int_{\mathbb{R}^n} \mathcal{H}^{N-n}(A\cap F^{-1}(y)) \mathrm{d}\mathcal{H}^n(y)$$ for all Lebesgue measurables $A\in\mathfrak{L}(\mathbb{R}^n)$ where $JF$ is the Jacobian $JF(x) := \det(DF(x)\cdot DF(x)^T)^{1/2}$. The change-of-variable formula is a special case of this and a combination of both shows the analogue formula for functions between Riemannian manifolds, the smooth coarea formula. Embedding this back into euclidean spaces this reads as $$\int_A J_D F(x) \mathrm{d}\mathcal{H}^D = \int_{\mathbb{R}^n} \mathcal{H}^{D-n}(A\cap F^{-1}(y)) \mathrm{d}\mathcal{H}^n(y)$$ for integers $n\leq D\leq N$ and subsets of embedded $D$-manifolds $A$ and a modified Jacobian $J_D F$.
My questions are simple: This last equation almost makes sense for all real numbers $D\in [n,N]$ and $\mathcal{H}^D$-measurable sets $A$. Is there an analogue of the Jacobian $J_D F$ that makes this fractional coarea formula true? If so, where can I find a proof?
(Similar questions have been asked, for example here (not answered) and here (answered, but not what I want to know) )