Low regularity theory for the inverse fractional conductivity problem

We characterize partial data uniqueness for the inverse fractional conductivity problem with $H^{s,n/s}$ regularity assumptions in all dimensions. This extends the earlier results for $H^{2s,\frac{n}{2s}}\cap H^s$ conductivities by Covi and the authors. We construct counterexamples to uniqueness on domains bounded in one direction whenever measurements are performed in disjoint open sets having positive distance to the domain. In particular, we provide counterexamples in the special cases $s \in (n/4,1)$, $n=2,3$, missing in the literature due to the earlier regularity conditions. We also give a new proof of the uniqueness result which is not based on the Runge approximation property. Our work can be seen as a fractional counterpart of Haberman's uniqueness theorem for the classical Calder\'on problem with $W^{1,n}$ conductivities when $n=3,4$. One motivation of this work is Brown's conjecture that uniqueness for the classical Calder\'on problem holds for $W^{1,n}$ conductivities also in dimensions $n \geq 5$.