Regularity of a function

Question

Let $U$ be a Banach Space. How regular is the $U$-valued function $f(t)$ defined as:

$$f(t)=\int_0^t g(t) dt$$

where $g\in L^p(0,1;U)$, $p>1$. We know that $f(t)$ is continuous in $t$, but is it Holder continuous with exponent $(p-1)/p$?