Stable extension of the EIT inverse map

By [F2] every conductivity has its unique zero-mean kernel. If ${\kappa }_{\gamma }={\kappa }_{{\gamma }_0}$, [F4] gives ${\dot{\mathrm{\Lambda }}}_{\gamma }={\dot{\mathrm{\Lambda }}}_{{\gamma }_0}$ and [F1], with its value zero at zero, gives $\gamma ={\gamma }_0$ in $L^2(\mathrm{\Omega })$. Thus assigning ${\kappa }_{\gamma }\mapsto \gamma$ defines a function on the exact image, independently of the conductivity representative. This remains valid if that image is empty.

For any exact pair, [F4] and [F1] give $\mathrm{\parallel }\gamma -{\gamma }_0{\mathrm{\parallel }}_2\le {\omega }_{\star }(\mathrm{\parallel }{\kappa }_{\gamma }-{\kappa }_{{\gamma }_0}{\mathrm{\parallel }}_2)$ with ${\omega }_{\star }(s)=C_0[\log (1\mathrm{/}\min \{C_1\sqrt{s},t_0\}){]}^{-\rho }$ for $s>0$ and value zero at zero. Its concavity and continuity are proved in [F3]. Applying that theorem to the function established in step 1.1 gives an extension to the entire underlying real Hilbert space of complex-valued kernels, with the same inequality for every pair of inputs and agreement on the exact image. If the class is empty, the zero map also supplies an extension. This proves both clauses.