# Change of Variable formula for a non-differentiable mapping.

Let $$\Omega \subset \Bbb R^n$$. For a diffeomorphism (or merely a differentiable bijection) $$\varphi:\Omega \to \varphi(\Omega)$$, we have the formula $$\int_{\Omega} f\circ\varphi^{-1}(x)\, dx = \int_{\varphi^{-1}(\Omega)} f(y)|D\varphi(y)| \,dy.$$

How much can we generalize the class in which $$\varphi$$ is allowed to lie in? Is it enough that we have, says, a bijection $$\varphi\in W_{\text{loc}}^{1,\infty}(\Omega ;\Bbb R^n)$$ or even $$\varphi\in W_{\text{loc}}^{1,1}(\Omega ;\Bbb R^n)$$?

How much does the result depends on the domain $$\Omega$$? Is there a big difference between a compact and an open domain? Does the regularity of the boundary $$\partial \Omega$$ play any role?

I'd also appreciate if you have a good reference to this kind of result so that I can read further into this interesting issue. Happy New Year to all of you.

Perhaps one of the most general classes of maps, defined on a set $$\Omega \subset \Bbb R^n$$, for which the change of variables formula $$\int\limits_{\Omega} f\circ\varphi^{-1}(x)\, \mathrm{d}x = \int\limits_{\varphi^{-1}(\Omega)} f(y)|D\varphi(y)| \,\mathrm{d}y \label{1}\tag{1}$$ (or a proper generalization) holds is the one considered by Piotr Hajłasz in [1]. To describe his results, it is useful to preliminarily recall some concepts.

• A function $$u:\Omega \to \Bbb R$$ is approximately totally differentiable at $$x_0\in\Omega$$ if there exists a real vector $$\mathsf{D}u|_{x_0}=(\mathsf{D}u_1,\ldots,\mathsf{D}u_n)$$ such that, for every $$\varepsilon$$, $$x_0$$ is a point of density for the set $$A_\varepsilon=\left\{ x\in\Bbb R\,\left|\;\frac{|u(x)-u(x_0)-\langle\mathsf{D}u|_{x_0},x-x_0\rangle|}{|x-x_0|}<\varepsilon\right.\right\}$$ Saying that $$u$$ is approximately totally differentiable or is approximately totally differentiable a.e. should have an obvious meaning.

• The class of approximately totally differentiable a.e. functions was characterized by Hassler Whitney in [2], pp. 144-147 (the statement of Whitney is slightly different though equivalent to the one reported in [1] pp. 93-94), by the following theorem 1: let $$u: E \to \Bbb R$$ be measurable, $$E \subseteq \Bbb R^n$$. Then the following conditions are equivalent:
(a) $$u$$ is approximately totally differentiable a.e. in $$E$$.
(b) $$u$$ is approximately derivable with respect to each variable a.e. in $$E$$.
(c) Denoting by $$|\cdot|$$ the Lebesgue measure, for each $$\varepsilon > 0$$ there exists a closed set $$F\subseteq E$$ and a function $$v\in C^1(\Bbb R^n)$$ such that $$|E\setminus F|<\varepsilon \text{ and }u|_F = v|_F.$$

• An approximately totally a.e. differentiable map $$\varphi:\Omega \to \varphi(\Omega)$$ is a map whose each component $$\varphi_i$$, $$i=1,\ldots, n$$ is approximately totally differentiable a.e. on its domain of definition $$\Omega$$.

• Let $$\varphi:\Omega \to \Bbb R^n$$. We say that $$\varphi$$ satisfies the condition N (Lusin’s condition) if for any $$E\subseteq\Omega$$, $$|E|=0 \implies |f(E)|=0.$$

• Let $$\varphi:\Omega \to \Bbb R^n$$, and $$E\subseteq\Omega$$. The Banach indicatrix is the function $$N_\varphi(\cdot ,E):\Bbb R^n\to \Bbb N\cup\{\infty\}$$ defined by $$N_\varphi(y, E) = \sharp(\varphi^{−1}(y) \cap E).$$ where $$\sharp$$ denotes cardinality measure of the given set.

After those preliminaries we can try to answer the OP questions:

How much can we generalize the class in which $$\varphi$$ is allowed to lie in?

It is a main result of [1] (Theorem 2, §2 pp. 94-96) ,that a generalization of formula \eqref{1} holds for the class of approximately totally a.e. differentiable maps.
Precisely, theorem 2 of [1] states that if $$\varphi:\Omega \to \Bbb R^n$$ is any mapping, where $$\Omega \subseteq \Bbb R^n$$ is an arbitrary open subset, satisfying one of the conditions (a), (b), (c), of theorem 1, then we can redefine it on a subset of measure zero in such a way that the new $$\varphi$$ satisfies the Lusin condition $$N$$.
If $$\varphi$$ satisfies one of the conditions (a), (b), (c) and the condition $$N$$, then for every measurable function $$f : \Bbb R^n \to \Bbb R$$ and every measurable subset $$E$$ of $$\Bbb R^n$$ the following statements are true:

1. The functions $$f(y)|D\varphi(y)|$$ and $$(f\circ\varphi^{-1}(x))N_\varphi(x, E)$$ are measurable.

2. If moreover $$f \ge 0$$ then $$\int\limits_E f(y)|D\varphi(y)|\mathrm{d}y = \int\limits_{\Bbb R^n} f\circ\varphi^{-1}(x)N_\varphi(x, E)\mathrm{d}x. \label{2}\tag{2}$$

3. If one of the functions $$f(y)|D\varphi(y)|$$ and $$(f\circ\varphi^{-1}(x))N_\varphi(x, E)$$ is integrable then so is the other (integrability of $$f |D\varphi|$$ concerns the set $$E$$) and the formula of \eqref{2} holds.

Note that

• Formula \eqref{2} is proved first for non-negative functions $$f\ge 0$$: the general case follows by the decomposition $$f= f^+ − f^−$$ ([1], §2 p. 96).

• I have modified the notation of [1] in order to show how formula \eqref{2} is a generalization of formula \eqref{1}, since this last one, proposed by the OP, has a non standard structure (even if it is perfectly equivalent to the standard one).

Is it enough that we have, says, a bijection $$\varphi\in W_{\text{loc}}^{1,\infty}(\Omega ;\Bbb R^n)$$ or even $$\varphi\in W_{\text{loc}}^{1,1}(\Omega ;\Bbb R^n)$$?

As recalled by Hajłasz ([1], example p. 94, and §3 p. 96), since the partial derivatives of $$\varphi\in W_{\text{loc}}^{1,1}(\Omega ;\Bbb R^n)$$ maps are defined a.e., these satisfy conditions (b) and (c) of theorem 1, implying that theorem 2 (and formula \eqref{2}) holds for them, so $$\varphi\in W_{\text{loc}}^{1,1}(\Omega ;\Bbb R^n)$$ is sufficient for the validity of formula \eqref{2}. Moreover Hajłasz ([1], §3 p. 96-98) is able to strengthen the theorem for these maps: however, this requires the same modification mechanism used in the general case, since there are continuous $$W_{\text{loc}}^{1,1}(\Omega ;\Bbb R^n)$$ maps which do not satisfy the Lusin's condition $$N$$.

How much does the result depends on the domain $$\Omega$$? Is there a big difference between a compact and an open domain? Does the regularity of the boundary $$\partial \Omega$$ play any role?

As you can see in the hypotheses of theorem 2, the domain $$\Omega$$ is only assumed to be an arbitrary open subset of $$\Bbb R^n$$ and it seems that its proof it does depend of the boundary structure (regularity) of the domain nor on its compactness (provided $$\Omega$$ has a non void interior, i.e. is compact in the sense that it has a compact closure). However, I do not have studied this paper carefully: perhaps I miss some subtleties of the proof which make my statement above imprecise/wrong.

[1] Piotr Hajłasz (1993), "Change of variables formula under minimal assumptions", Colloquium Mathematicum, 64, n. 1, pp. 93-101, ISSN 0010-1354; 1730-6302/e, DOI 10.4064/cm-64-1-93-101, MR1201446, Zbl 0840.26009.

[2] Hassler Whitney (1951), "On totally differentiable and smooth functions", Pacific Journal of Mathematics, Vol. 1 (1951), No. 1, 143–159, ISSN 0030-8730, DOI: 10.2140/pjm.1951.1.143, MR0043878, Zbl 0043.05803.

• Amazing answer! Thank you very much for such a detailed list of references. – BigbearZzz Jan 7 at 2:42
• @BigbearZzz: you are welcome. I'am happy to have been of some help. – Daniele Tampieri Jan 7 at 5:29