Question on Sobolev Space In learning the Sobolev space, I have a question why the Sobolev space $W^{k,p}$ could be embedded in the Holder space $C^{k,\alpha}$. Can we find a function in Holder space but not in the Sobolev space? And how is the norm in $C^{k,\alpha}$ connected with the norm in $W^{k,p}$? Though there is the Gagliardo-Nirenberg-Sobolev inequality and Morrey's inequality, I still could not understand in a simple way. Could anyone give some examples? The connectivity between the continuous function space  and $L^p$ function space is difficult to understand for me. 
In learning the graduate partial differential equations, I found many theorems abstract and difficult to understand, do you have any suggestion on how to get a direct and clear understanding of the theorems? Is there some book with examples to illustrates the fundamental theorems in partial differential equations like the sobole embedding theorem, the regularity theorems and the strong maximum principles. That you for your help!
 A: I don't know how much this will help, but:
The simplest instance of a Holder norm controlled by a Sobolev norm is that of the Fundamental Theorem of Calculus. Let us suppose we have a function $f:\mathbb{R}\to\mathbb{R}$ such that its distributional derivative $f'$ is defined, and such that we have $\int |f'|^p \mathrm{d}x < \infty$. Then we can estimate using the fundamental theorem of calculus:
$$ f(b) - f(a) = \int_a^b f'(x) \mathrm{d}x \leq (\int_a^b |f'|^p \mathrm{d}x)^{1/p} (\int_a^b 1 \mathrm{d}x)^{1/p*} $$
where the first inequality is Holder's inequality, and $p^*$ is the Holder conjugate of $p$ satisfying $1/p + 1/p^* = 1$. If $p = 1$ just take $1/p^* = 0$. The above expression implies
$$ |f(b) - f(a)| \leq  \|f'\|_{W^{1,p}} |b-a|^{1 - 1/p} $$
and so that $f$ must be Holder continuous with $\alpha = 1 - 1/p$. This is the simplest prototype of a Sobolev inequality. 
The proof of Morrey's inequality that I am familiar with factors through the Gagliardo-Nirenberg-Sobolev inequality. And if you look at the proof of that inequality, you will see that it is mostly just a fancy and optimal way of applying the fundamental theorem of calculus!
