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Let $X,Y$ be polish spaces, $s,t>0$ and $F:X\to Y$ locally Lipschitz continuous such that $X$ is $\sigma$-finite w.r.t. the $(s+t)$-dimensional Hausdorff measure $\mathcal{H}^{s+t}$.

The Eilenberg inequality shows $$\int_Y \int_{F^{-1}(y)} \chi_A(x) d\mathcal{H}^s(x) d\mathcal{H}^t(x) \leq Lip(F_{|A})^t \mathcal{H}^{s+t}(A)$$ for all $\mathcal{H}^{s+t}$-measurable $A\subseteq X$.

In particular the left hand side defines a Borel measure on $X$ which is absolutely continuous w.r.t. $\mathcal{H}^{s+t}$ so that there exists a measurable function $J^{s,t} F: X\to[0,\infty]$ with $$\int_Y \int_{F^{-1}(y)} \phi(x) d\mathcal{H}^s(x) d\mathcal{H}^t(x) = \int_X \phi(x) J^{s,t}F(x) d\mathcal{H}^{s+t}$$ for all measurable $\phi: X\to[0,\infty]$.

If $n\leq N$, $X\subseteq\mathbb{R}^n, Y=\mathbb{R}^N$ then the area formula shows $J^{0,n}F(x) = JF(x) = \det(DF(x)^T DF(x))^{1/2}$.

If $n\leq N$, $X\subseteq\mathbb{R}^N, Y=\mathbb{R}^n$ and $F$ is a $C^1$-submersion (or something sufficiently similar) then the coarea formula shows $J^{N-n,n}F(x) = JF(x) = \det(DF(x)DF(x)^T)^{1/2}$.

Question. Are there other cases of interest in which $J^{s,t}F$ is known or somehow "explicitly" definable from $F$?

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  • $\begingroup$ A very general Coarea formula for integer Hausdorff measure is proved in Federer's book, 3.2.22. Never heard for fractional ones. $\endgroup$ – Longyearbyen Dec 22 '16 at 8:03

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