Let $K\in D\{(x,\xi)\in \mathbb {R}^2 : x > 0, \xi > 0\} $ and $L (\phi)(x)=\int_0^x K (x,\xi)\phi (\xi)d\xi $ for $\phi \in D (\mathbb {R})$
$D $ is the space of testfunctions
I know that $\langle L\phi,\psi\rangle = \langle \phi, L^{\ast}\psi\rangle $ but how do i use it?
Hint: just write out the double integral which is $\langle L\phi,\psi\rangle$, apply Fubini and see what you get. $L^*$ is a form similar to the one of $L$ (only the integral kernel is slightly modified).