showing $W^{s,p}(\mathbb{R}^n) \subseteq W^{r,p}(\mathbb{R}^n)$ for $r < s$ I'm aware of a way of doing this using pseudo-differential operator theory.  One can easily reduce to showing that $W^{t,p}(\mathbb{R}^n) \subseteq L^p$ for $t > 0$.  This in turn follows because the identity map is a symbol operator of order zero, and hence also of order $t$, and so will map $W^{t,p}(\mathbb{R}^n)$ to $L^p$ (and also isomorphically to itself), by a general boundedness result for symbol operators.
I was wondering if anyone knows a more direct proof which doesn't rely on the general (and nontrivial) boundedness result for symbol operators.

You can find all you want here, in particular: Proposition 2.1, Proposition 2.2 and Corollary 2.3.
