> [!theorem] > > Let $U \subset \real^n$ be a bounded [[Open Set|open]] set, $f \in UC^1(U)$ be an [[Space of Uniformly Continuously Differentiable Functions|UC]] function, and $\varphi: \wh U \to U$ be a [$C^1$](Space%20of%20Continuously%20Differentiable%20Functions)-isomorphism where $\varphi, \varphi^{-1}$ are both $UC$. Then, there exists $C \ge 0$, independent of $f$, such that > $ > \norm{f \circ \varphi}_{W^{1, p}(\wh U)} \le C\norm{f}_{W^{1, p}(\wh U)} > $ > *Proof*. Firstly, over a change of variables, > $ > \begin{align*} > \int_{\wh U} \abs{f \circ \varphi}^p &= \int_U \abs{f}^p \cdot \abs{\det D\varphi} \\ > &\le \norm{\det D\varphi}_u \int \abs{f}^p \\ > \norm{f \circ \varphi}_p^p &\le \norm{\det D\varphi}_u \cdot \norm{f}_{W^{1, p}(U)}^p > \end{align*} > $ > On the other hand, $\abs{D(f \circ \varphi)(x)} = \abs{Df(\varphi(x)) \cdot D\varphi(x)}$, so > $ > \begin{align*} > \abs{D(f \circ \varphi)} &\le \abs{Df \circ \varphi} \cdot \norm{D\varphi}_u \\ > \int \abs{D(f \circ \varphi)}^p &\le \norm{D\varphi}_u^p \cdot \norm{\det D\varphi^{-1}}_u \cdot \norm{Df}_p^p > \end{align*} > $ > By passing over equivalent norms, there exists $C_n \ge 0$ such that $\norm{Df}_p \le \norm{f}_{W^{1, p}(U)}$. From here, > $ > \norm{D(f \circ \varphi)}_p^p \le C_n \cdot \norm{D\varphi}_u^p \cdot \norm{\det D\varphi}_u \cdot \norm{f}_{W^{1, p}(U)}^p > $ > Summing over all coordinates, we get that > $ > \norm{f \circ \varphi}_{W^{1, p}}^p \le \braks{nC_n \cdot \norm{D\varphi}_u^p \cdot \norm{\det D\varphi}_u + \norm{\det D\varphi}_u} \cdot \norm{f}_{W^{1, p}(U)} > $ > which is the desired bound.