Fix an integer value $k\geq 1$. Let $[0,1]$ the unit interval and let $s\in [0,1]$.
Does there exist a function $f$ (which depends on $k$ of course but not on $s$) such that $$\int_s^1 \left( \int_s^t f(u)du \right)^{-k} dt <\infty \quad \mbox{ for all } \quad s\in [0,1]\, ?$$
My first guess was to expect $t\mapsto \int_s^t f(u)du$ to be of order $(t-s)^{1/k+}$ but I don't find such an example. I might be blind? Thank you very much!