Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns to a prescribed boundary value $g$ a function $f$ with $f\rvert_{\partial\Omega}=g$ and $\Delta_\Omega f=0$.
I have two questions about this operator:
Can $E$ be bounded from $L^2(\partial \Omega)$ into $L^2(\Omega)$ as a right inverse of the trace operator? (or possibly another modification of $E$ or another extension operator).
Is there any explicit characterization of the range of $E$: $\mathrm{ran}(E):=E\,L^2(\partial \Omega)$?
Finally, any reference on some properties of such operator would be helpful.
