$\hat{T}: L_2[0,1] \to L_2[0,1]$ is a linear operator such that $(\hat{T}f)(x)=\int_0^1 \frac{f(t)}{|x-t|^{\frac{1}{4}}} dt$ with $x\in [0,1]$. I have to determine its norm. I've managed to show it's bounded using the Schwarz inequality: $$||\hat{T}f||^2=\int_0^1 \Big|\int_0^1 \frac{f(t)}{|x-t|^{\frac{1}{4}}} dt\Big|^2 dx=\int_0^1 |(k_x|f)|^2dx\leq ||f||^2\int_0^1 ||k_x||^2 dx=\frac{8}{3}||f||^2$$ where $k_x$ is the function $t\mapsto \frac{1}{|x-t|^{\frac{1}{4}}}$. Therefore $||\hat{T}||\leq\sqrt{\frac{8}{3}}$.
I know that the Schwarz inequality becomes an equality iff $k_x=\lambda f$ with $\lambda \in \mathbb{C}$, but I can't seem to find such a function for all $x$ and therefore I can't conclude that $||\hat{T}||=\sqrt{\frac{8}{3}}$ (I'm not even sure that is the norm actually). Any suggestion?