Iff conditions for a $C^1$-diffeomorphism to have $L^1$ or $L^\infty$ Jacobian

Question

Let $\Delta,D$ be two open subsets of $\mathbb{R}^d$, and let $\varphi:\Delta \rightarrow D$ be a $C^1$-diffeomorphism with Jacobian determinant $J_{\varphi}.$

1. Prove that $\lambda_d(D)<+\infty$ if and only if $J_{\varphi} \in L^1(\Delta).$

2. Prove that $J_\varphi$ is bounded on $\Delta$ if and only if $\exists c>0$ such that for all open $\Omega \subset\Delta$, $\lambda_d(\varphi(\Omega)) \leq c\lambda_d(\Omega).$

For part 1, the result follows from $\lambda_d(D)=\int_{\Delta}|J_{\varphi}(x)|dx.$

For part 2, if $J_\varphi$ is bounded, $\exists c>0$ such that for all open $\Omega \subset \Delta$,$$\lambda_d(\varphi(\Omega))=\int_{\Omega}|J_\varphi(x)|dx\leq c\lambda_d(\Omega).$$

How can we prove the converse?

Answer 1

Recall that for any continuous function $f$ defined on a neighbourhood of a point $x\in\mathbb R^d$, $$ \lim_{r\to 0}\frac{1}{\lambda_d(B(x,r))}\int_{B(x,r)}f(y) \, dy = f(x). $$

Suppose the continuous function $|J_\varphi|$ was unbounded. Then for each $n\in\mathbb Z_{>0}$, there exists $x_n\in \Delta$ such that $|J_\varphi(x_n)|>2n$. Therefore, for sufficiently small $r_n>0$, $$\frac1{\lambda_d(B(x_n,r_n))}\int_{B(x_n,r_n)}|J_\varphi(y)| \, dy > n,$$ which is to say $$ \lambda_d(\varphi( B(x_n,r_n) )) > n \lambda_d(B(x_n,r_n)),$$ so no such $c>0$ can exist.