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.