Let F be an absolutely continuous function on [0,1] and $F’\in L^p([0,1])$

Question

Let F be an absolutely continuous function on [0,1] and $F’\in L^p([0,1])$ For some $p\in (1,\infty)$ relative to Lebesgue measure. show that:

$\lim_{x\rightarrow0^+}(F(x)-F(0))x^{-1/q}=0$. Given $\frac{1}{p}+\frac{1}{q}=1$.

I’m thinking using Holder’s inequality to bound the term we are taking limit of because of the p q condition, and then the right side goes to zero. And $F(x)-F(0)=\int_0^xF’(t)dt$, but I have trouble continue here. Any help? Thanks!

Answer 1

Hint (along the lines of what you described, but more specific):

$$\left | \int_0^x F'(t) dt \right | x^{-1/q} \leq \| F' \|_{L^p([0,x])} \| 1 \|_{L^q([0,x])} x^{-1/q} \\ =\| F' \|_{L^p([0,x])}.$$