Why Do We Care About Hölder Continuity?

I have often encountered Hölder continuity in books on analysis, but the books I've read tend to pass over Hölder functions quickly, without developing applications. While the definition seems natural enough, it's not clear to me what we actually gain from knowing that a function is $\alpha$-Hölder continuous, for some $\alpha<1$.

I have some guesses, but they are just guesses: do $\alpha$-Hölder conditions give rise to useful weak solution concepts in PDEs? Are there important results that apply only to $\alpha$-Hölder functions, for some fixed $\alpha$? For $\alpha=1$ (Lipschitz continuity) the answer to both of these questions seems to be yes, but I know nothing for lower values of $\alpha$.

I'd be interested in answers that describe specific applications, as well as answers that give more of a ''big picture''.

Hölder continuous functions do not give rise to useful weak solutions in any context I am aware of: there are notions of weak solutions that are continuous, but the Hölder modulus is not relevant for the definition.

While there may be some rare results that require specific Hölder moduli with $\alpha < 1$, I can not think of any that I use in my research.

So why care about Hölder continuity at all? Here are a few reasons. I will say that this is coming from a purely PDE perspective, and that Hölder spaces are at their most useful when dealing with elliptic, parabolic, and some first-order PDE. For dispersive and wave equations, the fact that Hölder norms do not interact well with the Fourier transform is a strike against them. There are other (non-PDE) areas of analysis and geometry that find Hölder spaces useful for other reasons, but that would be for another answer.

Compactness

Hölder spaces have very elementary and favorable compactness properties. A sequence of functions with bounded Hölder norms will have a uniformly convergent subsequence, and the Hölder norm is lower semicontinuous under uniform convergence. Uniform convergence is extremely, surprisingly, useful when studying some types of PDE, and is often enough to pass the entire PDE to the limit. This is the case with distributional solutions of linear equations, and more strikingly with viscosity solutions.

Easy to use and understand

The theory of Hölder spaces is not very deep. Unlike Sobolev spaces, which interact in subtle ways with the geometry of a domain's boundary, contain functions that generally don't make sense pointwise, require dealing with distributional derivatives, etc., Hölder spaces are just spaces of equicontinuous functions with little else going on.

It is easy to prove that a function is Hölder continuous, and common ways of doing so line up well with the way we approach PDE. One way to do this is to prove that $$ \max_{B_r(x)} u - \min_{B_r(x)} u \leq (1 - \theta)[ \max_{B_{2r}(x)} u - \min_{B_{2r}(x)}u] $$ for some $\theta > 0$, a decay of oscillation. Iterating this gives that $u$ has some Hölder modulus at $x$. This kind of statement is one we are happy to try and prove for solutions $u$: bounding the maximum of $u$ at a given scale in terms of $u$ on a larger ball is something we actually have the tools to do, at least for elliptic equations. There are also good approaches to showing $u$ is Hölder based on Sobolev or $L^p$ bounds at every scale (Morrey/Companato inequalities), and sometimes Sobolev spaces embed into Hölder spaces directly.

Another good aspect of Hölder spaces is that they let us talk about a fractional-power increase in smoothness without having to take (fractional?) derivatives, or any derivatives at all, and without needing the Fourier transform. Not having to take derivatives is a great technical convenience (see how the improvement of oscillation above is a statement about just the solution pointwise; this is great if differentiating the equation is problematic); not having to deal with anything fractional makes everything much more explicit; not needing the Fourier transform is good news for equations that interact poorly with it.

Our best theorems are true when $\alpha \in (0, 1)$

Sure, Hölder spaces might be nice, but why not just use $\alpha = 1$? It turns out that it is much, much harder to prove something is Lipschitz, and that often it is just not true. Consider the equation $$ \Delta u = f. $$ Heuristically we expect that $u$ is two derivatives smoother than $f$, because, well, that's what the equation seems to say: some second derivatives of $u$ equal $f$. The actual positive results in this direction are that if $f \in C^{0, \alpha}$, then $u \in C^{2, \alpha}$ (Schauder), that if $f \in L^p$ then $u \in W^{2, p}$ when $p \in (1, \infty)$ (Calderon-Zygmund), some similar theorems that are $k$ derivatives up from this, and much more complicated classifications of what happens at the endpoints $\alpha = 0, \alpha = 1, p = 1, p = \infty$. In particular, none of the endpoint versions are true, they all require modifications, different spaces, etc. This fact, that harmonic analysis theorems have more complicated endpoint versions, is a running theme in the field, and means that while we would love to work with $\alpha = 1$, often we just are not allowed to.

There are other types of theorems where we can prove that there exists an $\alpha > 0$ such that solutions (or their derivatives, or something related to them) are in $C^{0, \alpha}$. Here we may not be expecting the functions to be anywhere near Lipschitz. The most famous example of this is the De Giorgi-Nash-Moser theorem.