Hilbert-valued extension with a concave modulus

Consider the set of pairs $(B,h)$ where $A\subseteq B\subseteq H_1$ and $h:B\to H_2$ extends $g$ and satisfies $\mathrm{\parallel }h(u)-h(v)\mathrm{\parallel }\le {\omega }_{\star }(\mathrm{\parallel }u-v\mathrm{\parallel })$ for all $u,v\in B$. Order it by graph inclusion. It is nonempty, since $(A,g)$ is in it. Every nonempty chain has as upper bound the union of its graphs: every two of its domain points occur together in one member of the chain, so the same modulus inequality holds for their union. The empty chain has $(A,g)$ as an upper bound. By [F4] and [A1] there is a maximal pair $(B,h)$.

For the EIT application first arrange $t_0\le e^{-(\rho +1)}$. If the original IP-45 constants have a larger cutoff, decrease it and enlarge $C_0$ if needed: the uniform class diameter is at most $2M\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}2}$, so taking $C_0\ge 2M\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}2}[\log (1\mathrm{/}{t}_0){]}^{\rho }$ covers all DN distances above the new cutoff, while increasing $C_0$ preserves the old small-distance estimate. For $0<s<(t_0\mathrm{/}{C}_1{)}^2$, put $L(s)=\log (1\mathrm{/}(C_1\sqrt{s}))$ and $\phi (s)=C_{0}L(s{)}^{-\rho }$. Then ${\phi }^{\mathrm{\prime }}(s)=C_0\rho \mathrm{/}(2sL(s{)}^{\rho +1})>0$ and ${\phi }^{\mathrm{\prime }\mathrm{\prime }}(s)=C_0\rho [\rho +1-2L(s)]\mathrm{/}(4s^{2}L(s{)}^{\rho +2})\le 0$. At the cutoff it joins continuously to a constant and its derivative drops to zero; at zero it tends to zero. Thus the displayed ${\omega }_{\star }$ is continuous, nondecreasing and concave on the entire nonnegative half-line.

If $B\mathrm{\ne }{H}_1$, fix $x\in H_1\setminus B$. Apply [F2] to the source family $(u{)}_{u\in B}$ and target family $(h(u){)}_{u\in B}$. It gives a point $y\in H_2$ satisfying $\mathrm{\parallel }y-h(u)\mathrm{\parallel }\le {\omega }_{\star }(\mathrm{\parallel }x-u\mathrm{\parallel })$ for every $u\in B$. Adding $h(x)=y$ gives a strictly larger admissible graph, contradicting maximality. This also covers $B=\mathrm{\varnothing }$, since [F2]'s empty intersection is $H_2$, which contains zero. Thus $B=H_1$ and $\tilde{g}=h$ proves the full same-modulus extension theorem for arbitrary real Hilbert spaces and arbitrary subsets.

For an exact conductivity pair put $s=\mathrm{\parallel }{\kappa }_{\gamma }-{\kappa }_{{\gamma }_0}{\mathrm{\parallel }}_{L^2(M\times M;\mathbb{C})}$. By [F1], [F3] and monotonicity, $\mathrm{\parallel }\gamma -{\gamma }_0{\mathrm{\parallel }}_2\le C_0[\log (1\mathrm{/}\min \{C_1\sqrt{s},t_0\}){]}^{-\rho }={\omega }_{\star }(s)$ when $s>0$; if $s=0$, [F3] gives zero DN distance and [F1] gives equal conductivities. The operators ${\mathcal{R}}_{\gamma }$ and $C_M^{r_d\mathrm{/}2}$ in the smoothed observation commute with complex conjugation: the weak Neumann problem has real coefficients and unique mean-zero representatives, while the real boundary Laplacian has conjugation-invariant eigenspaces and the spectral multiplier is real and constant on each eigenspace. Thus ${\tilde{A}}_{\gamma }$ commutes with conjugation. Let ${\kappa }_{\gamma }$ be its unique complex $L^2$ kernel from the Hilbert–Schmidt kernel theorem. The kernel $\overline{{\kappa }_{\gamma }}$ also represents ${\tilde{A}}_{\gamma }$, since its integral operator sends $f$ to $\overline{{\stackrel{~}{A}}_{\gamma }\stackrel{‾}{f}}={\tilde{A}}_{\gamma }f$. Uniqueness of the kernel gives $\overline{{\kappa }_{\gamma }}={\kappa }_{\gamma }$ almost everywhere. In particular, the exact kernel image lies in the real-valued subspace, while it is still regarded as a subset of the underlying real Hilbert space of complex-valued kernels. Therefore the exact-data inverse is well-defined and has this modulus on that real Hilbert space. Step 2.1 applies there and in the real conductivity $L^2$ Hilbert space, with concavity verified in step 1.2, to give its whole-space extension with exactly the same modulus.