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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!

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  • $\begingroup$ Firstly, pay attention to the exponents. In general $W^{k,p}$ does not embed in $C^{k,\alpha}$. You must lose derivatives. Secondly, the Sobolev space does not "embed" per say. What is shown is the a priori estimate that for $C^\infty_0$ functions that the $C^{k,\alpha}$ norm can be controlled by the $W^{k,p}$ norm. The subtle distinction is that $W^{k,p}$ functions are equivalent classes of functions agreeing almost everywhere. You can easily change a $W^{k,p}$ function at a single point, so that it is no longer continuous, without changing its norm. $\endgroup$ – Willie Wong Nov 20 '13 at 9:21
  • $\begingroup$ If one wants to really split hairs, what one can say is that every $W^{k,p}$ function with suitably high $k$ has, in its equivalent class, a function that is in $C^{k',\alpha}$ for suitable $k',\alpha$. What you need to get used to is that in mathematics, a lot of times "equivalence classes" of objects are just treated as if there is only one object, with a specific representative. $\endgroup$ – Willie Wong Nov 20 '13 at 9:23
  • $\begingroup$ I understand your words. but I still could not understand whycould the $C^{k,\alpha}$ norm controlled by the $W^{k,p}$ norm? Is there some direct insight or pictures, or some examples on this? $\endgroup$ – sam Nov 20 '13 at 9:30
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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!

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