Fundamental Theorem of Calculus for distributions. Consider a function $F \in C^{\alpha}( \mathbb{R})$ for $0 < \alpha < 1.$ Then we can take it's distributional derivative. We can say $f = F' \in C^{\alpha -1}( \mathbb{R} )$. My issue is going back.
Say I have a $\alpha -1 -$Hölder function $f$, then how can I "integrate" it to get a primitive $F$ such that $f$ is it's distributional derivative?
Here we use the following definition (but any other definition can be used as well) for negative $\beta \in (-1,0)$: $$g \in C^{\beta}( \mathbb{R} ) \Leftrightarrow \ |\langle g, \phi_{x}^{\lambda} \rangle| \  ≤ C \lambda^{\beta}$$ where $\phi_{x}^{\lambda}(z) = \lambda^{-d}\phi(\frac{z -x}{\lambda})$ and $C$ is uniform over all $x$ and $\phi \in C^{\infty}_0(-1,1)$ with $||\phi||_{C^1}≤1.$
EDIT:


*

*An approach would be to prove that $$C^{\beta} \subseteq L^1_{Loc}$$ which would mean in particular that our distributions are actually functions. This does not work, as pointed out in the comments, and in related questions.

*My other idea was to define $\langle f, 1_{[0,a]} \rangle $ by using typical approximation of the indicator function of $1_{[0,a]}.$ This cannot work: for example the derivative of a the Brownian motion is not a measure.
I saw that this topic is covered only slightly in MathStackExchange. Here are similar questions regarding this topic.


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*Here Is a general question about Holder spaces with negative exponents.

*Here Is a question that is very similar to mine, maybe just in a slightly different context.

*Here Is a question by myself regarding the same topic. In fact as you might imagine I am looking into this subject with little success :D


Finally let me also give some motivation as to why I ma studying this: these spaces are used in the theory of SPDEs. In particular a reference is Fritz's and Hairer's book: "A course on rough paths." Both my questions are exercises in this book.
 A: I don't have the book you mention, so I don't know what tools you are allowed
to use to solve those exercises, but anyway. First you need to show that your
distribution $g$ has order one. Take $\psi\in C_{c}^{\infty}(\mathbb{R})$ with
support in some interval $K=[-n,n]$ and $\Vert\psi\Vert_{C^{1}}\leq1$. Then
$\phi(x):=\psi(nx)$, $x\in\mathbb{R}$, has support in $[-1,1]$ and so by
applying the bound with $\lambda=n$ and $x=0$ you get that $\phi_{0}%
^{n}(x)=n^{-1}\phi(x/n)=n^{-1}\psi(x)$. Hence, by linearity,
$$
|\langle g,\psi\rangle|=n|\langle g,n^{-1}\psi\rangle|=n|\langle g,\phi
_{0}^{n}\rangle|\leq n(Cn^{\beta}).
$$
Now for every $\psi\in C_{c}^{\infty}(\mathbb{R})$ with support in $K$,
applying the previous inequality with $\psi$ replaced by $\psi/\Vert\psi
\Vert_{C^{1}}$, you get, again by linearity,
$$
|\langle g,\psi\rangle|\leq Cn^{1+\beta}\Vert\psi\Vert_{C^{1}}=C_{K}\Vert
\psi\Vert_{C^{1}}.
$$
This shows that $g$ has order one. Now any distribution can be written as sum
of derivatives of continuous functions and if the order is $m$ you only need
$m+2$ functions (you can find this in Edward's functional, or Rudin's
functional analysis, or Leoni's Sobolev books). 
So in your case you can find $f_{0}$, $f_{1}$, $f_{2}$ and $f_{3}$ continuous
such that
$$
\langle g,\psi\rangle=\int_{\mathbb{R}}f_{0}\psi-\int_{\mathbb{R}}f_{1}%
\psi^{\prime}+\int_{\mathbb{R}}f_{2}\psi^{\prime\prime}-\int_{\mathbb{R}}%
f_{3}\psi^{\prime\prime\prime}.
$$
Define $F_i(x):=\int_{0}^{x}f_i(t)\,dt$ for $i=0,1,2$ and
$$\langle G,\psi\rangle=\int_{\mathbb{R}}F_{0}\psi-\int_{\mathbb{R}}F_{1}%
\psi^{\prime}+\int_{\mathbb{R}}F_{2}\psi^{\prime\prime}-\int_{\mathbb{R}}%
F_{3}\psi^{\prime\prime\prime}.$$
Then integrating by parts
\begin{align}\langle G^\prime,\psi\rangle=-\langle G,\psi^\prime\rangle
&=-\int_{\mathbb{R}}F_{0}\psi^\prime+\int_{\mathbb{R}}F_{1}%
\psi^{\prime\prime}-\int_{\mathbb{R}}F_{2}\psi^{\prime\prime\prime}+\int_{\mathbb{R}}%
F_{3}\psi^{\prime\prime\prime\prime}\\
&=\int_{\mathbb{R}}f_{0}\psi-\int_{\mathbb{R}}f_{1}%
\psi^{\prime}+\int_{\mathbb{R}}f_{2}\psi^{\prime\prime}-\int_{\mathbb{R}}%
f_{3}\psi^{\prime\prime\prime}
\end{align}
There are probably simpler proofs done by hand (probably the proof that a
distribution can be written as sum of derivatives has all the tricks you need
to find a primitive). Mollifying $g$ might also do the trick.
