Mean-zero NtD map and smoothed kernel observation

Assume the Axiom of Choice and Countable Choice. Let $d\ge 2$, let $\mathrm{\Omega }\subset (0,1{)}^d$ be nonempty, bounded, connected and $C^{\mathrm{\infty }}$, fix an open domain ${\mathrm{\Omega }}^{\mathrm{\prime }}⋐\mathrm{\Omega }$ and real parameters $m>0$, $M\ge \max \{1,m\}$, $R>0$ and $0<\alpha <1$ defining $X_d(R)$, and let $\gamma \in X_d(R)$ be a real conductivity. Write $M=\mathrm{\partial }\mathrm{\Omega }$, let $dS$ be surface measure, and set $L_0^2(M)=\{f\in L^2(M):{\int }_{M}fdS=0\},P_{0}f=f-\mathrm{\mid }M{\mathrm{\mid }}^{-1}{\int }_{M}fdS\mathrm{.}$ The mean-zero Neumann-to-Dirichlet operator ${\mathcal{R}}_{\gamma }:L_0^2(M)\to L_0^2(M)$ sends $f$ to the mean-zero boundary trace of the weak solution of $\mathrm{\nabla }\cdot (\gamma \mathrm{\nabla }u)=0$ with outward conormal flux $f$, as defined by the weak Neumann problem. Let $-{\mathrm{\Delta }}_M$ be the nonnegative Laplace–Beltrami operator on the compact boundary manifold, with eigenpairs $(e_j,{\lambda }_j)$ on $L_0^2(M)$ from its spectral resolution. For $\tau =1\mathrm{/}2$ when $d=2$ and any fixed $\tau >0$ when $d\ge 3$, put $r_d:=\frac{d-1}{2}-1+\tau ,C_M^{r_d\mathrm{/}2}{e}_j=(1+{\lambda }_j{)}^{-r_d\mathrm{/}2}{e}_j\mathrm{.}$ The smoothed observation is the bounded operator $A_{\gamma }:={\mathcal{R}}_{\gamma }{C}_M^{r_d\mathrm{/}2}:L_0^2(M)\to L_0^2(M)$. Its extension to all of $L^2(M)$ is ${\tilde{A}}_{\gamma }f={\iota }_0{A}_{\gamma }(P_{0}f)$, where ${\iota }_0:L_0^2(M)\hookrightarrow L^2(M)$ is inclusion. Thus its integral kernel, when it exists in $L^2(M\times M)$, is fixed to have zero mean in each variable. No assertion of a kernel's existence or Hilbert–Schmidt regularity is part of this definition.

The dimension-dependent conductivity class.

Existence and uniqueness of the weak mean-zero Neumann solution.

The boundary spectral resolution and Sobolev scale.

The boundary surface measure and its mean.

The weak energy-form Dirichlet-to-Neumann convention.

Well-posedness and quotient conventions for the weak DN map.