Krdytkyu

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

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

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
$$
intlimits_{Omega} fcircvarphi^{-1}(x), mathrm{d}x = intlimits_{varphi^{-1}(Omega)} f(y)|Dvarphi(y)| ,mathrm{d}y
label{1}tag{1}
$$
(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 $x_0inOmega$ 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
  $$
  A_varepsilon=left{ xinBbb R,left|;frac{|u(x)-u(x_0)-langlemathsf{D}u|_{x_0},x-x_0rangle|}{|x-x_0|}<varepsilonright.right}
  $$
  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 $Fsubseteq E$ and a function $vin C^1(Bbb R^n)$ such that
  $$
  |Esetminus F|<varepsilon text{ and }u|_F = v|_F.
  $$




• 
  

  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 $EsubseteqOmega$,
  $$
  |E|=0 implies |f(E)|=0.
  $$




• Let $varphi:Omega to Bbb R^n$, and $EsubseteqOmega$. The Banach indicatrix is the function $N_varphi(cdot ,E):Bbb R^nto Bbb Ncup{infty}$ 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 $f(y)|Dvarphi(y)|$ and $(fcircvarphi^{-1}(x))N_varphi(x, E)$ are measurable.




2. If moreover $f ge 0$ then
   $$
   intlimits_E f(y)|Dvarphi(y)|mathrm{d}y = intlimits_{Bbb R^n}
   fcircvarphi^{-1}(x)N_varphi(x, E)mathrm{d}x. label{2}tag{2}
   $$



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

Note that

• Formula eqref{2} is proved first for non-negative functions $fge 0$: 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 $varphiin W_{text{loc}}^{1,infty}(Omega ;Bbb R^n)$ or even $varphiin W_{text{loc}}^{1,1}(Omega ;Bbb R^n)$?

As recalled by Hajłasz ([1], example p. 94, and §3 p. 96), since the partial derivatives of $varphiin W_{text{loc}}^{1,1}(Omega ;Bbb R^n)$ 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 $varphiin W_{text{loc}}^{1,1}(Omega ;Bbb R^n)$ 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.