Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 8 / 5 / i / Solution

For $u\in C_c^{\infty }({\mathbb{R}}^d)$, the fundamental theorem of calculus along the $j$th coordinate line givesTake $f=|u|$ and $s_j=|{\partial }_{j}u|$ in the introductory result. It follows thatThe last step is the arithmetic-geometric mean inequality. This bounds the target norm by the first-order Sobolev space norm. Approximate any $u\in W^{1,1}({\mathbb{R}}^d)=L_1^1({\mathbb{R}}^d)$ by compactly supported smooth functions, using cutoffs and mollification as in Question 4. The estimate applied to differences gives convergence in $L^{d/(d-1)}$, and subsequences of the simultaneous $L^1$ and target-norm convergence identify the two limits almost everywhere. Thus $W^{1,1}$ embeds continuously into $L^{d/(d-1)}$. The coordinate-derivative product bound also passes to the limit. For $d=1$, the line argument instead gives $\Vert u{\Vert }_{\infty }\le \Vert u^{\prime }{\Vert }_1$ and the embedding into $L^{\infty }$.