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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 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 might contain something you can use.

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.

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 what's available in the literature concerning bounds for the operator norm of $\varphi_1^*- \varphi_2^*$, but you might want to take a look at The Pullback Equation for Differential Forms by Csato, Dacorogna and Kneuss in case there are any regularity results in there you can use.

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 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 might contain something you can use.

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 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 what's available in the literature concerning bounds for the operator norm of $\varphi_1^*- \varphi_2^*$, but you might want to take a look at The Pullback Equation for Differential Forms by Csato, Dacorogna and Kneuss in case there are any regularity results in there you can use.

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 what's available in the literature concerning bounds for the operator norm of $\varphi_1^*- \varphi_2^*$, but you might want to take a look at The Pullback Equation for Differential Forms by Csato, Dacorogna and Kneuss in case there are any regularity results in there you can use.