Fundamental Theorem of Calculus for distributions.

Question

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:

1. 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.

2. 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.

1. Here Is a general question about Holder spaces with negative exponents.
2. Here Is a question that is very similar to mine, maybe just in a slightly different context.
3. 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.

Answer 1

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.