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This is mainly concerned with studies coming from Gilbarg and Trudinger. Elliptic Partial differential equations are of the form $Lu =f$, where $L = a^{ij}D_{ij} + b^iD_i + cu$, and most of the results come assuming that $f \in C^{\alpha}$, that is, $f$ is holder continuous, $0<\alpha<1$. What the book doesn't really explain is why we consider holder continuous functions, and not some other class of functions.. for example $C^0, C^1$, etc. I get that the estimates work for $C^{\alpha}$, but "because it works" isn't really a good explanation.

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The Holder space $C^{\alpha}(X)$ is stronger than $C(X)$ but weaker than $C^1(X)$. Overall, it is simply natural in the analysis of function spaces to consider such spaces when you're trying to figure out how "nice" a function is. The Holder seminorm for $0 < \alpha < 1$ $$ \|f\|_{C^{\alpha}} = \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^{\alpha}}$$ measures how much the function $f$ behaves somewhat like how $x^{\alpha}$ behaves near $x = 0$. Certainly a function like $x^{1/2}$ can be considered somewhat "nice" near $x = 0$ even though it fails to have a derivative at that point.

From a PDE standpoint, Morrey's inequality tells us that Sobolev spaces can be embedded into a certain Holder space, which again tells us how "nice" Sobolev spaces can end up being. In the context of your book, Gilbarg and Trudinger, proving regularity results for $C^{\alpha}$ is "sharper" than proving regularity in other more traditional spaces like $C^k$ with $k \in \mathbb{Z}^{+}$, because it tells us that we don't actually need functions to be that nice.

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I like the answer, but I think it's kind of what I was trying to avoid. The book makes estimates for $C^{\alpha}$ functions... because that's what we can make nice estimates for. I'm not sure then if there really is a good answer for the question. Certainly I agree that proving results for $C^{\alpha}$ is "sharper" than $C^{k}$, but it just seems quite arbitrary to me to introduce the class of functions without motivation as to why we are choosing these functions to study. –  Euler....IS_ALIVE Oct 6 '12 at 23:20
    
Well, I think at least the first half of my answer has some intuitive motivation; the idea is that studying differentiable functions implies that we require our functions to have tangent lines, so functions can be approximated by lines. A $C^{\alpha}$ function can only be approximated by sublinear functions, namely functions that look like $x^{\alpha}$. –  Christopher A. Wong Oct 6 '12 at 23:50
    
Yes that's true. I would also be curious about whether Morrey's inequality inspired the study of Holder spaces, or if it was the other way around, or if it was independent! –  Euler....IS_ALIVE Oct 6 '12 at 23:54
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