I'm working my way through some old analysis quals at my university and I came across this question. Let $f$ be absolutely continuous on $[0,1]$ with $f(0)=0$ and $f'\in L^3([0,1])$. For which values of $\alpha$ does $$ \lim_{x\rightarrow 0^+} x^{-\alpha}f(x)=0$$ for all such $f$?
I have tried the following approach. Using the FTOC for Lebesgue integrals and Holder's inequality: $x^{-\alpha}f(x)=\int_0^xf'(t)x^{-\alpha}dt\leq ||\chi_{[0,x]}|f'(t)|^3||_{3,[0,1]}x^{-\alpha/3}$. The quanitity on the right hand side will go to zero if $\alpha>0$. So this doens't seem to be super helpful. How should I proceed?