= Solution
For $u\in C_c^\infty(\mathbb R^d)$, the <fundamental theorem of calculus> along the $j$th coordinate line gives
$$
|u(x)|\leq\int_{\mathbb R}|\partial_ju(x_1,\ldots,x_{j-1},t,x_{j+1},\ldots,x_d)|\,dt.
$$
Take $f=|u|$ and $s_j=|\partial_ju|$ in the introductory result. It follows that
$$
\boxed{\|u\|_{d/(d-1)}\leq\prod_{j=1}^d\|\partial_ju\|_1^{1/d}\leq\frac1d\sum_{j=1}^d\|\partial_ju\|_1}.
$$
The 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. \b[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 $\|u\|_\infty\leq\|u'\|_1$ and the embedding into $L^\infty$.
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