= Solution
For an integer $s\geq0$, the <Sobolev space> is
$$
W^{s,p}(\mathbb R^d)=\{u\in L^p:D^\beta u\in L^p\text{ for every multi-index }|\beta|\leq s\},
$$
where $D^\beta$ is a <weak derivative>. For $p<\infty$ one may use the <Sobolev norm> $(\sum_{|\beta|\leq s}\|D^\beta u\|_p^p)^{1/p}$; for $p=\infty$ use the maximum of the finitely many essential-supremum norms.
For $1\leq p<d$, the <Sobolev inequality> is $\|u\|_{p^*}\leq C_{d,p}\|\nabla u\|_p$, with <Sobolev conjugate exponent> $p^*=dp/(d-p)$. For $d<p<\infty$, <Morrey's inequality> supplies a continuous representative satisfying
$$
|u(x)-u(y)|\leq C_{d,p}|x-y|^{1-d/p}\|\nabla u\|_p,
\qquad
\|u\|_\infty\leq C_{d,p}(\|u\|_p+\|\nabla u\|_p).
$$
At $p=\infty$ the representative is <Lipschitz continuous>. At the critical exponent $p=d>1$, first-order Sobolev regularity gives every finite $L^q$ embedding for $q\geq d$, with an inhomogeneous norm, but generally no $L^\infty$ embedding. The one-dimensional endpoint $W^{1,1}(\mathbb R)\hookrightarrow L^\infty(\mathbb R)$ is an exception.
For the proof of <Morrey's inequality>, start with a smooth $u$ and write $u_{B(x,r)}$ for its average on a ball. Averaging the <fundamental theorem of calculus along a line segment> and changing radial variables gives
$$
|u(x)-u_{B(x,r)}|
\leq C_d\int_{B(x,r)}\frac{|\nabla u(z)|}{|x-z|^{d-1}}\,dz
\leq C_{d,p}r^{1-d/p}\|\nabla u\|_p.
$$
The last step is the <Holder inequality>; integrability of the kernel to power $p'$ is exactly $(d-1)p'<d$. For $r=|x-y|$, translate the averaging ball along the segment from $x$ to $y$. The <fundamental theorem of calculus along a line segment> and the <Holder inequality> give
$$
|u_{B(x,r)}-u_{B(y,r)}|\leq r|B_r|^{-1/p}\|\nabla u\|_p.
$$
Combining the two point-to-average bounds and this average-to-average bound proves the required Hölder estimate. The point-to-average bound with $r=1$, together with $|u_{B(x,1)}|\leq|B_1|^{-1/p}\|u\|_p$, gives the supremum estimate. <Density of smooth functions in a Sobolev space> then gives a uniformly convergent sequence of smooth representatives, preserving both bounds. For $p=\infty$, mollification gives the Lipschitz version.
For the decay conclusion assume $d<p<\infty$. The representative is uniformly continuous. If $|u(x_j)|\geq2\theta>0$ along points escaping to infinity, the Hölder bound gives a radius $\rho>0$, independent of $j$, on which $|u|\geq\theta$. A subsequence has disjoint radius-$\rho$ balls, each contributing at least $\theta^p|B_\rho|$ to $\|u\|_p^p$, a contradiction. This is <uniformly continuous integrable functions vanish at infinity>.
\b[The finite-$p$ restriction is necessary.] If the printed range includes $p=\infty$, its decay assertion is false: $u\equiv1$ belongs to $W^{1,\infty}$ but does not tend to zero.
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