Fractional Calder\'on problem with $L^{\infty}$-conductivities

Abstract

We study the inverse problem for the fractional conductivity equation with uniformly elliptic conductivities $\gamma\in L^\infty(\mathbb R^n)$. For every $n\geq1$ and $0<s<1$, we prove that the local exterior Dirichlet-to-Neumann map on any nonempty open subset of the exterior of a bounded open domain determines $\gamma$ uniquely almost everywhere in $\mathbb R^n$. No continuity, Sobolev regularity, or prescribed behavior at infinity is required. The proof works directly with the conductivity bilinear form and avoids the fractional Liouville reduction. It combines exterior determination at Lebesgue points, equality of weighted solutions, and a unique continuation principle for the fractional Laplacian in the weighted class \[ L_s=\left\{ g\in L^1_{\mathrm{loc}}(\mathbb R^n): \int_{\mathbb R^n} \frac{|g(x)|}{1+|x|^{n+2s}}\,dx<\infty \right\}. \] Specifically, if $g\in L_s$ and both $g$ and $(-\Delta)^s g$ vanish in a nonempty open set, with the latter equality understood distributionally, then $g=0$ almost everywhere. This result requires neither local boundedness nor Sobolev regularity of $g$. Its proof uses the Caffarelli--Silvestre extension and boundary unique continuation for the associated degenerate elliptic equation.