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\newcommand{\c}[1]{\mathcal{C}^{#1}(\Omega)}
$So I've been struggling with this for a while, and I keep getting what feels like a lot of conflicting answers regarding these when I search these up, so I would like to ask you guys for some clarification on these terms — both formal and intuitive ideas if possible.
I recognize this might feel a little trivial to you guys and quite haphazard but between personal searching and what feels like conflicting ideas between two different classes intended to be rife with this theory, the searches, and textbooks, I'm just quite confused and am probably getting many wires crossed. Hopefully you can help me figure out what exactly it all means.
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Suppose $f \in \mathcal{C}^{\alpha}(\Omega)$. It is clear what this means to me when $\alpha$ is an integer. What does it mean when it is not an integer? For instance, what if $f \in \mathcal{C}^{2+\beta}(\Omega)$ for $\beta \in (0,1)$?
- One definition given I recall is that a function is $\c\alpha$ if its derivatives up to the $\alpha$th order (all such mixed derivatives included in the multivariate case). Another, attempting to handle the fractional case (e.g. the $\c{2+\beta}$ case above) suggested that we use multi-indices … but since those lie in $\Bbb Z^n$ it feels like it's no different a characterization in the first place, so there's clearly a missing link there.
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Suppose $f \in \mathcal{C}^{\alpha,\beta}(\Omega)$: notice how we use two superscripts now. What does this mean?
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One suggestion I came across was the claim that it mean it was $\c \alpha$ in a first variable and $\c\beta$ in the second, but I'm looking at functions that are not explicitly in terms of any number of variables, so this does not feel like a correct characterization. Especially as I've only seen at most two superscripts so far.
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Another is essentially that $\c{\alpha,\beta}$ is all $\c \alpha$ functions with all derivatives finite under the Hölder seminorm of exponent $\beta$ on all $\Omega' \subseteq \Omega$ of compact closure.
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A third is that (taking $k,\ell$ integers, $\alpha,\beta\in(0,1)$) we have $\c{k+\alpha,\ell+\beta}$ all those $\c{k,\ell}$ functions $f$ whose $k$th derivative $\mathcal{D}^k f$ finite under the Hölder seminorm $[\cdot]_{\alpha,\beta}$.
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These spaces are evidently Banach spaces over the respective field. What exactly is the norm on these spaces, with the one- or two-superscript case? How are fractional such indices handled?
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At least the Hölder seminorm I find clear agreement on. For exponent $\mu$ and domain $\Omega$, the Hölder seminorm is
$$[f]_{\mu,\Omega} = \sup_{\substack{x,y \in \Omega \\ x \ne y}} \frac{|f(x)-f(y)|}{|x-y|^\mu}$$ -
One thing brought up in my notes however references two indexing subscripts; for $\alpha \in (0,1)$:
$$[f]_{\alpha,\alpha/2} = \sup_{\substack{(x,t),(y,s) \in \Omega \times (0,t) \\ (x,t) \ne (y,s)}} \frac{|u(x,t) – u(y,s)|}{|x-y|^\alpha – |t-s|^{\alpha/2}}$$
This was called the Hölder coefficient of $f$ of exponent $\alpha$. Isn't the Hölder coefficient what is used to bound $|g(x)-g(y)|$ in the ordinary definition? (Though I suppose we could probably use the same thing we do in functional analysis with norms and prove them to be the same…) What is this all about? This was soon followed by mentioning the Hölder spaces $\c{2+\alpha,1+\alpha/2}$ and $\c{\alpha,\alpha/2}$ in no real detail. -
From here, we brought up the proper norm,
$$\| f \|_{\c{\alpha,\alpha/2}} := \| f \|_{\c k} + [f]_{\alpha,\alpha/2}$$
No clue what this $k$ is about. Should be an $\alpha$? -
Another suggestion noted was having
$$\|u\|_{\c{2+\alpha,1+\alpha/2}} = [u]_{\alpha,\alpha/2} + \sup_{\Omega} \left( \sum_{i=0}^2 \mathcal{D}^i u(x,t) \right)$$
alongside
$$\|u\|_{\c{2,1}} = \sup_{\Omega} \left( \sum_i \left| \mathcal{D}^i u \right| \right)$$
(summing over all partial derivatives as necessary) so I guess for fractional indices you effectively only use what you would for integers, and then from there tack on a seminorm tied to the fractional indices?
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So, summarily:
- What, formally, is a $\c \alpha$ function for $\alpha$ a noninteger?
- What, formally, is a $\c{\alpha,\beta}$ function, particularly in the noninteger cases?
- What, precisely defined, are these norms and seminorms we wish to deal with in either case?
Thanks for any insights you can give, and sorry for being so readily frazzled.
Best Answer
After digging through some texts and poking my professor, I came to know that the main text he has been relying on so far is Nonlinear Parabolic & Elliptic Equations, by C.V. Pao (Amazon link). This text was able to finally give me some firmer grounding as to what (most of) the terms mean.
So, I'll cover the basics needed in full generality and then define the terms.
Notation & Conventions: Throughout, we assume the following and define the following notations:
$ \newcommand{\o}{\Omega} \newcommand{\ol}{\overline} \newcommand{\a}{\alpha} \newcommand{\CC}{\mathcal{C}} \newcommand{\c}[1]{\CC^{#1}} \newcommand{\R}{\mathbb{R}} \newcommand{\p}{\partial} \newcommand{\set}[1]{\left\{ #1 \right\}} \newcommand{\setb}[2]{\left\{ #1 \, \middle| \, #2 \right\}} \newcommand{\n}[2]{\left\| #1 \right\|_{\CC^{#2}(\o)}} \newcommand{\abs}[1]{\left| #1 \right|} \newcommand{\Z}{\mathbb{Z}} \newcommand{\DD}{\mathcal{D}} \newcommand{\N}{\mathbb{N}} \newcommand{\nd}[2]{\left\| #1 \right\|_{\CC^{#2}(D_T)}} $ $\o \subseteq \R^n$ is a our domain in $\R^n$; it may be bounded or unbounded, closed or open, unless stated otherwise. Of course, $\p\o$ will denote its boundary, and $\ol\o$ its topological closure. We will refer to $\o$ as being the spatial coordinate(s).
Let $T > 0$; then we define $D_T := \o \times (0,T]$, $S_T := \p \o \times (0,T]$. $(0,T]$ will generally represent the temporal coordinate.
Unless stated otherwise, $\alpha$ is a constant and $\alpha \in (0,1)$.
We take the convention that $0 \in \N$, i.e. $\N = \Z_{\ge 0}$.
Definition ($\CC^k$ functions & analogues): We define, for $k,\ell,m \in \Bbb Z_{\ge 1}$,
$$\begin{align*} \CC(\o) &:= \setb{ f : \o \to \R }{ \text{$f$ is continuous} } \\ \c k(\o) &:= \setb{ f \in \CC(\o) }{ \text{all partial derivatives of $f$ up to order $k$ are $\CC(\o)$ functions}} \\ \c{\ell,m}(D_T) &:= \setb{ f : D_T \to \R }{ \text{$f$ is $\CC^\ell$ spatially, and $f$ is $\CC^m$ temporally } } \end{align*}$$
(Where $\c{\ell,m}$ is concerned, note that this does in fact coincide with being $\CC^\ell$ in the first variable and $\CC^m$ in the second. Hence, for instance, if $f \in \c{2,1}(\o \times (0,\infty))$, then $f$ is twice continuously differentiable in space, and once in time. That is, if $f(x,t)$ is the convention for naming the variables, then $f$ is twice continuously differentiable in $x$, and once in $t$.)
Analogous notations exist for considerations of the closures of $\o,D_T,$ or other sets in a more general context. We sometimes may think of $\CC(\o)$ as being $\CC^0(\o)$.
Definition ($\CC$ norm): For $f \in \CC(\o)$, we define its norm in the usual way, via supremum:
$$\n{f}{} := \sup_{x \in \o} \abs{ f(x) }$$
Per usual, if we change the set involved, analogous notations arise.
Definition ($\a$-Hölder continuity): We say $f \in \CC(\o)$ is $\a$-Hölder continuous, or Hölder continuous with exponent or order $\a$, if
$$H_\alpha(f) := \sup_{\substack{x,y \in \o \\ x \ne y}} \frac{\abs{ f(x) - f(y) }}{ \abs{x-y}^\a } < \infty$$
($H_\a$ gives the Hölder seminorm. I usually see it denoted just $[f]_{\a}$ but this feels easier for me to parse.) We may equivalently state that $f$ is $\a$-Hölder continuous if $\exists k > 0$ such that, $\forall x ,y \in \o$,
$$\abs{ f(x) - f(y) } \le k \cdot \abs{x-y}^\a$$
Definition (Hölder norm): The Hölder norm (w.r.t. $\a$) of $f \in \CC(\o)$ is given by
$$\n{f}{\a} = \n{f}{} + H_\a(f)$$
Definition ($\c \a$ functions): We define now
$$\c \a(\o) := \setb{ f \in \CC(\o) }{ \n{f}{\a} < \infty }$$
Definition (Multi-Indices & Derivative Notation): An $n$-tuple $k := (k_i)_{i=1}^n \in \N^n$ is said to be a multi-index, and its order $\abs{k}$ is given by
$$\abs{k} = \sum_{i=1}^n k_i$$
We then say that, if differentiable, a function $f$ has derivative $\DD^k f$ where that is taken to mean
$$\DD^k f = \frac{\p^{|k|}}{\p_{x_1}^{k_1} \p_{x_2}^{k_2} \cdots \p_{x_n}^{k_n}} f$$
for that given multi-index $k$. Finally, by convention, we say that $\DD^k f = f$ for any multi-index $k$ of order $0$ (or rather, $k = (0,0,\cdots,0)$).
Definitions ($\c{k}$ & $\c{k+\a}$ norms): We have already seen the norm on $\c\a(\o)$,
$$\n{f}{\a} := \n{f}{} + H_\a(f)$$
where
$$\n{f}{} := \sup_{x \in \o} \abs{ f(x) } \qquad H_\a(f) := \sup_{\substack{x,y \in \o \\ x \ne y}} \frac{\abs{ f(x) - f(y) }}{ \abs{x-y}^\a }$$
We may generalize this norm further. We can find the $\CC^k$ norm of $f$ by summing up the $\CC^0$ norms of it and all partial derivatives up to order $k$, i.e.
$$\begin{align*} \n{f}{k} &:= \n{f}{} + \sum_{\substack{\ell \in \N^n \\ \abs{\ell} = 1}} \n{\DD^\ell f}{} + \sum_{\substack{\ell \in \N^n \\ \abs{\ell} = 2}} \n{\DD^\ell f}{} + \cdots + \sum_{\substack{\ell \in \N^n \\ \abs{\ell} = k}} \n{\DD^\ell f}{} \\ &= \n{f}{} + \sum_{m = 1}^k \sum_{\substack{\ell \in \N^n \\ \abs{\ell} = m}} \n{\DD^\ell f}{} \end{align*} $$
In a similar spirit, we may define the norms for $\CC^{k+\a}$ functions, by considering the $\CC^\a$ norms where the derivatives are concerned, i.e.
$$\n{f}{k+\a} := \n{f}{} + \sum_{m = 1}^k \sum_{\substack{\ell \in \N^n \\ \abs{\ell} = m}} \n{\DD^\ell f}{\a} $$
Definition ($\c{k+\a}$ function space): With the above established, we consequently define
$$\c{k+\a}(\o) := \setb{ f \in \CC^k(\o) }{ \n{f}{k+\a} < \infty }$$
Remark: $\CC^{k+\a}(\o)$ is Banach w.r.t. these norms for $k=0,1,2$.
Definition (Spacetime Hölder Constant): We also would like to introduce the Hölder constant for the case where we have a time variable, i.e. when we're dealing with $D_T$ as opposed to merely $\o$. We define it by
$$H_{\a,\a/2}(f) := [f]_{\a,\a/2} := \sup_{\substack{(x,t),(y,s) \in D_T \\ (x,t) \ne (y,s)}} \frac{ \abs{ f(x,t) - f(y,s) } }{ \abs{x-y}^{\a} + \abs{t-s}^{\a/2} }$$
Note/Concern: In the textbook mentioned previously, a slightly different definition is given (below). The above is the one given by my professors; perhaps they're equivalent or at least there is no functional difference? The definition per Pao is
$$H_{\a,\a/2}(f) := [f]_{\a,\a/2} := \sup_{\substack{(x,t),(y,s) \in D_T \\ (x,t) \ne (y,s)}} \frac{ \abs{ f(x,t) - f(y,s) } }{ \left( \abs{x-y}^2 + \abs{t-s} \right)^{\a/2} }$$
Definition (Spacetime Hölder Norm): Consequently, we define the Hölder norm for a function of domain $D_T$ to be
$$\nd{f}{\a} := \n{f}{} + H_{\a,\a/2}(f)$$
and in turn
$$\CC^{\a}(D_T) := \setb{ f \in \CC(D_T) }{ \nd{f}{\a} < \infty }$$
Similarly, we take (when $k \ge 1$)
$$\nd{f}{k+\a} :=\nd{f}{} + \nd{\DD_t f}{\a} + \sum_{m = 1}^k \sum_{\substack{\ell \in \N^n \\ \abs{\ell} = m}} \nd{\DD_x^\ell f}{\a}$$
where $\DD_x$ signifies the derivatives being done spatially.
We then define
$$\CC^{k+\a}(D_T) := \setb{ f \in \CC^k(D_T) }{ \nd{f}{k+\a} < \infty }$$
From here I don't have explicit confirmation for the remaining concerns (namely about spaces of the form $\CC^{k+\alpha,\ell+\beta}$ for $\alpha,\beta \in (0,1)$ -- e.g. what is $\CC^{2+\a,1+\a/2}$ and its norm?) -- but extrapolating the pattern above at least helps me feel more sure about the suggestions given.