ca.analysis and odes - variational formulation: boundedness of the bilinear form

The simplest case of the problem I'm thinking about involves an elliptic differential operator, $Lu = -u'' + qu$, on the interval $(0,1)$, with homogeneous Dirichlet boundary conditions. I want to show that the bilinear form on $H_0^1 subset H_1$ defined by

$a(u,v) = int_0^1 u'v' + quv~dx$

is bounded for the $H^1$-norm, i.e., $|a(u,v)| leq M|u|_1|v|_1$ for some constant $M>0$.

My question: can I assume that the linear coefficient $q$ is $L^1$ or even $L^2$ and still guarantee boundedness?

I was thinking that this is possible, but the only books that I have lying around discussing this consider only the case when $q$ is smooth or $L^infty$. I've played around with the Cauchy-Schwartz inequality for the term $int quv$ but am not getting anywhere.

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