The Rellich-Kondrachov theorem says that the inclusion of $W^{1,p}(U)$ in $L^q(U)$ is a compact operator for $q < dp/(d-p)$, where U is a bounded open subset of $d$ dimensional Euclidean space with $C^1$ boundary. In light of the Riesz-Kolmogorov theorem this implies that if $T_a$ is the translation operator by some vector $a$, then we should be able to find a function $\alpha$ such that for each $f \in W^{1,p}(U)$,
$$ \| T_a f – f \|_{L^q} \leq \alpha(a) \| f \|_{W^{1,p}}, $$
where $\alpha(a)$ converges to zero as $a \to 0$. What methods enable us to prove this directly?
Best Answer
Note that if $p=q$, this follows easily from taking a derivative: $$(T_af-f)(x)=\int_0^a{(\nabla u)(x+y)\,dy}$$ By Minkowski's inequality, we can move the $p$-norm inside the integral, so that $$\|T_af-f\|_p\leq\int_0^a{\|T_y\nabla u\|_p\,dy}$$ Since $T_y$ is norm-preserving, we are integrating a constant. Thus $$\|T_af-f\|_p\leq a\|\nabla u\|_p$$
For the general case, one can use the Sobolev inequalities, although this may not be as elementary as you like: $$\|T_af-f\|_{L^q}\leq C\|T_af-f\|_{W^p}\leq2C\|f\|_{W^p}$$ by triangle inequality.