## 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 $r$, but 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 to me, 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?
1. 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?
1. 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.