Hölder continuity of integral function

Question

Let $[a,b] \subset \mathbb{R}$ and $p \in [1, \infty)$. Consider $f \in L^{p}([a,b])$ with respect to the Lebesgue measure and set $F(x) = \int_{a}^{x} f(t) dt $ $\ \ \forall x \in [a,b].$

Prove that $F$ is Hölder continuous in $[a,b]$.

Can anyone give me a hint for the proof? I can observe that $f \in AC([a,b])$ and $f \in BV([a,b])$. Is this going to help me with the proof? Thanks everyone in advance.

Answer 1

Hint : note that $F(x) - F(y) = \int_{[x,y]} f(s)\text{d}s = \int \mathbb{1}_{[x,y]} \cdot f(s) \text{d}s$ then invoke Hölder's inequality.