Prove $\Lambda(f)(x)=\int_{0}^{1}\frac{f(t)}{\sqrt{|x-t|}}dt$ is bounded operator from $L^2[0,1]$ to $L^2[0,1]$
My idea was to use Jensens but it does not seem to work.
$$\int_{0}^{1}{\left(\int_{0}^{1}\frac{f(t)}{\sqrt{|x-t|}}dt\right)}^2dx\leq \int_{0}^{1}{\int_{0}^{1}\frac{f(t)^2}{|x-t|}}dtdx=\int_{0}^{1}f(t)^2\ln\left(1-\frac{1}{t}\right)dt$$ But the logarithm is not in $L^\infty$ and so i cannot use holders and finish the proof. I used Fubini and Jensen in above.
How do I solve this problem? There is a hint that says to use $\sqrt{|x-t|}=\sqrt[4]{|x-t|}\sqrt[4]{|x-t|}$ but I do not see how.