Linearized Calderón Problem: Reconstruction and Lipschitz Stability for Infinite-Dimensional Spaces of Unbounded Perturbations

We investigate a linearised Calderón problem in a two-dimensional bounded simply connected C1,α domain Ω. After extending the linearised problem for L²(Ω) perturbations, we orthogonally decompose L²(Ω) = ⊕ₖ₌₀^∞ Hₖ and prove Lipschitz stability on each of the infinite-dimensional Hₖ subspaces. In particular, H₀ is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearised) Calderón problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmic-type in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert-Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fréchet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general L²(Ω) perturbation onto the Hₖ spaces, hence reconstructing any L²(Ω) perturbation.