Upper bound on Lp distance of functions before and after change of variables

Question

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

1. Are there well-known results of such an upper bound under some mild conditions?
2. 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?
3. 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.

Answer 1

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.