I need a local Sobolev regularity result for a smooth solution $u$ of $$ -\Delta u=0 $$ with the equation satisfied in an open set $U$ (I have no boundary conditions). I know that such a smooth $u$ exists, and I have a potentially useful estimate on $u$ in $H^{-2}(U)$, i.e. the dual space of $H^2_0(U)$, which I want to bootstrap to $L^2(U)$. In other words, I want to be able to claim that $$ \|u\|_{L^2}\le c\|u\|_{H^{-2}} $$ (with a constant that doesn't depend on $u$). Is this possible?
If the answer is yes, I would like to have a specific reference that gives this result without requiring too much "translation" from the general to the particular (at present I have an indication that this is true with a pointer to Chapter 7 of Morrey's book on multiple integrals in the calculus of variations, but this is not convincing).
Any help would be much appreciated.
James
The answer is no. Let $\,U\subset\mathbb{R}^n\,$ be some bounded domain satisfying the cone condition, $n\geqslant 2$, i.e., the embedding $\,L^2(U)\hookrightarrow H^{-2}(U)\,$ is compact. The assumption that an estimate $$ \|u\|_{L^2(U)}\leqslant c\|u\|_{H^{-2}(U)}\tag{$\ast$} $$ can hold for all harmonic $\,u\in L^2(U)\,$ with some constant $c>0$ independent of $u$ would imply that the subspace $\{u\in L^2(U)\,\colon\, \Delta u=0\,\}$ be locally compact and hence finite-dimensional, which is definitely wrong. That is why $\,(\ast)\,$ cannot hold for all harmonic $\,u\in L^2(U)\,$ with a constant independent of $\,u$.