Upper bound on Lp distance of functions before and after change of variables Setup
I am trying to upper-bound the difference between two functions: one before the change of variables and the other after.
For example, let $r \in \mathbb{N} \cup \{\infty\}, 1 \leq p < \infty$ and
$$
g \in L^p(\mathbb{R}^D) : C^r \text{-function}\\
T : \mathbb{R}^D \to \mathbb{R}^D : C^r\text{-function} \\
f_T(x) := g(T(x)) \left|\det \frac{\mathrm{d} T}{\mathrm{d}x}(x)\right|.
$$
Goal
I would like to bound the perturbation due to $T$, in terms of $L^p$ norm.
Let
$$
F[T] := \|f_T - g\|_{L^p} := \left(\int |f_T (x)  - g(x)|^p dx\right)^{1/p}.
$$
For two $C^r(\mathbb{R}^D, \mathbb{R}^D)$-functions, $T, T'$, I would like to have an upper-bound on the difference in $F$, i.e.,
$$
F[T'] - F[T] \leq (\text{Some upper bound depending on } (T' - T)).
$$
The result does not have to be for all $p$ and $r$, and I am interested in any result of this type for a specific set of  $p$ or $r$. The space where these functions reside in is also variable, and any appropriate suggestion for change is very appreciated.
Questions


*

*Are there well-known results of such an upper bound under some mild conditions?

*Is the above example an appropriate setup? Are there suggestions such as what function space to use or what properties to assume on the functions?

*If not, does anybody have any idea how I should derive an upper bound?


P.S. I'm not fully familiar with functional analysis, so if there are any ambiguities or imprecise terminology, I will fix it if you could point them out.
 A: This isn't an answer, but it's a bit long for a comment. I am going to write $\varphi$ for your $T$, just for consistency with the notation in a book I'm going to mention.
Let $X = \{ g \hspace{.2pc} \mathrm{d}x^1\wedge \dots \wedge \mathrm{d}x^D: g\in L^p(\mathbb{R}^D)\}$. Then each suitable$_1$ diffeomorphism $\varphi$ of $\mathbb{R}^D$ gives us an associated pullback operator $\varphi^*:X \to X$ which, assuming you stick to orientation-preserving $\varphi$, is just the linear map $g\mapsto f_T$. The quantity you're interested in is then
$$
F[\varphi] = \Vert (I-\varphi^*)g\Vert,
$$
where $I:X\to X$ is the identity operator. You then have:
$$
\begin{array}{lll}
F[\varphi_1] & = & \Vert (I-\varphi_1^*)g\Vert \\
&=& \Vert ( I-\varphi_2^* + \varphi_2^* - \varphi_1^*)g\Vert \\
&\leq& F[\varphi_2] + \Vert (\varphi_2^* - \varphi_1^*)g\Vert
\end{array}
$$
whence $F[\varphi_1] - F[\varphi_2] \leq \Vert g \Vert\Vert \varphi_1^*-\varphi_2^*\Vert_\mathrm{op}$ for any pair of orientation-preserving $\varphi_1$ and $\varphi_2$.
I am not sure what bounds are available for the operator norm of $\varphi_1^*- \varphi_2^*$, but The Pullback Equation for Differential Forms by Csato, Dacorogna and Kneuss seems like it might contain some related material.
