Periodic $H^1((0,2\pi)),L^2((0,2\pi)),L^1((0,2\pi))$

Question

Let $a\in L^1((0,2\pi))$,$u,v\in H^1((0,2\pi))$ such that : $u(0)=u(2\pi),v(0)=v(2\pi)$.

Can we prove that : $$\int_{0}^{2\pi}(a(x))^2u(x)v(x)dx<\infty, ?$$

I really don't know this is right or wrong .