Bounded variation after a diffeomorphism

Question

This might be a standard property of BV functions but I have not heard about it before.

Let $I=[0,1]$ and $\phi:I\to \mathbb R$. We say that $\phi$ is of bounded (total) variation if $$ \sup_{ 0=x_0<\ldots < x_k < \ldots < x_n = 1 } \sum_{k=0}^{n-1} |\phi(x_{k+1}) - \phi(x_k)| < \infty .$$

Questions: is it true that

• if $\phi$ is of bounded variation then for every $C^1$-diffeomorphism $p:I\to I$ the function $\phi\circ p$ is of bounded variation;
• if there exists some $C^1$-diffeomorphism $p:I\to I$ such that the function $\phi\circ p$ is of bounded variation, then $\phi$ is of bounded variation?

The same questions for $p\circ \phi$ (probably they are identical).

Answer 1

1. If $p\colon[0,1]\to[0,1]$ is Lipschitz (in particular, if it is a $C^1$ diffeomorphism) with Lipschitz constant $L$, then $$ |p\circ \phi(x_{k+1})-p\circ \phi(x_{k})|\le L|\phi(x_{k+1})-\phi(x_{k})|. $$ If $\phi$ is BV, so is $p\circ \phi$.

2. Let $p\colon[0,1]\to[0,1]$ be strictly increasing. If $\{x_k\}_{k=0}^n$ is a partition of $I$, then $\{p(x_k)\}_{k=0}^n$ is also partition of $I$ (after adding the point $x=1$ and removing repeated points if necessary.) Then $$ \sum|\phi\circ p(x_{k+1})-\phi\circ p(x_k)|\le\|\phi\|_{BV}. $$

3. If $p$ is a $C^1$ diffeomorphism so is $p^{-1}$.