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Let $p\in(1,\infty)$ and $\beta:=1-\frac{1}{p}$ and define $Tf:[0,1]\rightarrow \mathbb R$, $x\mapsto \int_{[0,x]}fd\lambda$. Show:
$A_p:=\{\alpha \in (0,1): Tf\text{ Hölder continuous for the exponent $\alpha$ for every $f\in L^p([0,1])$}\}=(0,\beta]$.
Hint: Show $(0,\beta]\subseteq A_p$ and show $(\beta,1)\subseteq A_p^C$. Look at the functions $t\mapsto t^\gamma$ for $\gamma \in \mathbb R$ and choose $\|f\|_p$ as the Hölder constant.
I dont have a idea to prove this. Thanks for a hint. For the first inclusion I tried to show $|\int_{[0,x]}-\int_{[0,y]}|\leq \|f\|_p|x-y|^\alpha$ for $\alpha \in (0,\beta]\subseteq (0,1)$. I don't know how to simplify the expression $|\int_{[0,x]}-\int_{[0,y]}|$

physicsStudent
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  • Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Another User Jun 10 '22 at 15:44
  • Take a look at the answer here it's the same trick. – Arctic Char Jun 10 '22 at 15:57

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