## Sobolev or L^p for a pde proofs

when I look at the equation say $u_t=Lu=a(x,t)u_x+b(x,t)u_{xx}$ $a$ and $b$ are smooth functions. Now look at a particular scheme I want to prove the stability of this scheme, say it is Crank-Nicolson. If I pick the L^2 space, then I have little hope to prove results because L doesn't have to map into L^2, correct? Because first and second derivative don't have to be square integrable. So does it mean I have to start always with a smaller range of functions i.e. at least H^2? thanks!

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