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I've started working with Holder spaces recently and I'm wondering how I should think of them intuitively? I really have no idea what a function $f$ that is Holder continuous with exponent $\alpha$ is supposed to look like whereas I do have a good idea for other function spaces.

$L^{\infty}$: Say we had a function $f$ such that $\Vert f \Vert_{L^\infty([0, 2])} \le 1$. Then I can visualize $f$ as being some function in the box $[0, 2] \times [-1, 1]$.

Lipschitz: If we take another function $g$ and say it is Lipschitz continuous with some fixed constant $C$ I know that the slope of the $g$ will always be less than $C$ at any point in its domain which agrees with the vizualization stated on the Wikipedia page regarding a double cone that can be translated along the graph such that the graph always remains outside the cone.

Holder: What is the best way to visualize a Holder continuous function with exponent $\alpha$ on a given domain such as, say, $[0, 5]$? Does such a function have a clear geometric interpretation like $L^{\infty}$ functions? What would be an example of such a function if the exponent was $\alpha = 0.2$ for example? Would a function with an exponent of $\alpha = 0.3$ be 'nicer' than one with $\alpha= 0.2$?

csss
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Hoelder continuity is about the roughness of a path. So there are some extremes. First of all if $f$ is $\alpha$ Hoelder continuous with $\alpha>1$, then $f$ is constant (very easy to prove).

A function that is Hoelder continuous with $\alpha=1$ is differentiable a.e.

So if you're Hoelder continuous with $\alpha\ge 1$ things are very nice. Less than $1$ and things are much less nice.

The lower your Hoelder exponent is, the rougher the path is. In particular $\alpha=\frac12$ is very critical. I do research in rough path theory, which handles the case when $\alpha<\frac12$ but this is much more advanced.

I think the best way to understand different Hoelder continuity is to look at some paths! I do work with what's called fractional Brownian motion. Fractional Brownian motion is a stochastic process that has a parameter $H$. $H$ ends up being the Hoelder exponent of the path a.s.

So look at some fractional Brownian motions! Here's a few pictures from Wikipedia that I think will really help clarify how "rough" a path is.

Also to answer your question, would a Hoelder continuous path with $\alpha=.3$ be nicer than $.2$, YES.

  • Great answer, I now understand precisely what I'm dealing with when I have function quantified by its Holder exponent! Do you have any experience with Holder continuous functions in relation to PDEs? Why would we want to use Holder spaces instead of Lp spaces or Sobolev spaces when dealing with PDEs? – csss Dec 07 '16 at 15:07
  • @csss I have a lot of experience with Hoelder continuous functions in relation to PDEs. As a matter of fact, the entire reason why I care about Hoelder continuity is because of (stochastic) PDEs, my research area. Let me just state a fact: the Young integral $\int_0^T f~dX$, where $X,f$ are Hoelder continuous paths with exponent $\alpha, \beta$ respectively, converges iff $\alpha+\beta>1$. What does this have to do with PDEs? Well quite a bit actually. Let me write down a so called controlled differential equation: $dY=f~dX$. What is the obvious solution? Still not a PDE, but motivation. –  Dec 07 '16 at 17:04
  • Do you have proof of your statement "A function that is Holder continuous with $\alpha=1$ is differentiable a.e."? – user48672 May 19 '17 at 21:25
  • @user48672 https://math.stackexchange.com/questions/2101956/lipschitz-continuity-implies-differentiability-almost-everywhere –  May 19 '17 at 22:24