We investigate a linearised Calder\'on problem in a two-dimensional bounded simply connected $C^{1,\alpha}$ domain $\Omega$. After extending the linearised problem for $L^2(\Omega)$ perturbations, we orthogonally decompose $L^2(\Omega) = \oplus_{k=0}^\infty \mathcal{H}_k$ and prove Lipschitz stability on each of the infinite-dimensional $\mathcal{H}_k$ subspaces. In particular, $\mathcal{H}_0$ 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) Calder\'on 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 Fr\'echet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general $L^2(\Omega)$ perturbation onto the $\mathcal{H}_k$ spaces, hence reconstructing any $L^2(\Omega)$ perturbation.

翻译：我们研究了二维有界单连通$C^{1,\alpha}$域$\Omega$中的线性化Calderón问题。将线性化问题推广至$L^2(\Omega)$扰动后，我们对$L^2(\Omega) = \oplus_{k=0}^\infty \mathcal{H}_k$进行正交分解，并证明了在每个无穷维子空间$\mathcal{H}_k$上的Lipschitz稳定性。特别地，$\mathcal{H}_0$是平方可积调和扰动空间。这似乎是(线性化)Calderón问题中关于无穷维扰动空间的首个Lipschitz稳定性结果。在此之前，无穷维框架下关于数据映射算子范数的最优估计均为对数型。这一显著改进得益于采用Hilbert-Schmidt范数处理Neumann-to-Dirichlet边界映射及其关于电导率系数的Fréchet导数。我们还提出了一种直接重构方法，可归纳性地得到一般$L^2(\Omega)$扰动在$\mathcal{H}_k$空间上的正交投影，从而实现对任意$L^2(\Omega)$扰动的重构。