The translation estimate for $W^{1,p}$ functions on ${\mathbb{R}}^n$

Given: the Axiom of Choice, $n\ge 1$, $1\le p<\mathrm{\infty }$, a scalar field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, a class $u\in W^{1,p}({\mathbb{R}}^n;\mathbb{K})$ and a vector $h\in {\mathbb{R}}^n$.

Smooth density in $W^{1,p}({\mathbb{R}}^n)$. Under Countable Choice, for every $u\in W^{1,p}({\mathbb{R}}^n;\mathbb{K})$ there are ${\phi }_m\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{K})$ with $\mathrm{\parallel }{\phi }_m-u{\mathrm{\parallel }}_{W^{1,p}}\to 0$; Countable Choice is supplied by the assumed Axiom of Choice. (Compactly supported smooth functions are dense in W^{k,p}(R^n), The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))

Fundamental theorem of calculus. For a smooth function $\phi$, $\phi (x-h)-\phi (x)=-{\int }_0^{1}D\phi (x-th)\cdot hdt$, with the complex-valued identity read componentwise. (Fundamental theorem of calculus for absolutely continuous functions)

Minkowski and translation isometry. Minkowski's integral inequality applies to the $t$-integral on $[0,1]$, and Lebesgue translation invariance gives $\mathrm{\parallel }w(\cdot -th){\mathrm{\parallel }}_p=\mathrm{\parallel }w{\mathrm{\parallel }}_p$ for every $w\in L^p$. (Minkowski's integral inequality, Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, The space $L^p(\mu )$ as the quotient by null functions)

Vector gradient norm. The $L^p$ norm of the Euclidean magnitude $\mathrm{\mid }Du\mathrm{\mid }=({\sum }_i\mathrm{\mid }{D}_{i}u{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$ is equivalent, with constants depending only on $n,p$, to the finite sum of the component $L^p$ norms used in Integer-order Sobolev spaces and their norms. Indeed $\mathrm{\mid }{D}_{i}u\mathrm{\mid }\le \mathrm{\mid }Du\mathrm{\mid }\le {\sum }_i\mathrm{\mid }{D}_{i}u\mathrm{\mid }$, so Minkowski bounds $\mathrm{\parallel }\mathrm{\mid }Du\mathrm{\mid }{\mathrm{\parallel }}_p$ above by ${\sum }_i\mathrm{\parallel }{D}_{i}u{\mathrm{\parallel }}_p$. Also $\mid \mathrm{\mid }D{u}_m\mathrm{\mid }-\mathrm{\mid }Du\mathrm{\mid }\mid \le \mathrm{\mid }D(u_m-u)\mathrm{\mid }\le {\sum }_i\mathrm{\mid }{D}_i(u_m-u)\mathrm{\mid }$, which proves convergence in $L^p$ when $u_m\to u$ in $W^{1,p}$. (Integer-order Sobolev spaces and their norms)

Weak derivatives. A locally integrable function $v$ has weak ${\mathrm{\partial }}_j$-derivative $w$ when $\int v{\mathrm{\partial }}_j\psi =-\int w\psi$ for every $\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$; if $v,w\in L^p$, this places $v$ in $W^{1,p}$ in that coordinate. (Weak derivative of a locally integrable function, Integer-order Sobolev spaces and their norms)