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We generalize many recent uniqueness results on the fractional Calder\'onproblem to cover the cases of all domains with nonempty exterior. The highlightof our work is the characterization of uniqueness and nonuniqueness of partialdata inverse problems for the fractional conductivity equation on domains thatare bounded in one direction for conductivities supported in the wholeEuclidean space and decaying to a constant background conductivity at infinity.We generalize the uniqueness proof for the fractional Calder\'on problem byGhosh, Salo and Uhlmann to a general abstract setting in order to use the fullstrength of their argument. This allows us to observe that there are alsouniqueness results for many inverse problems for higher order localperturbations of a lower order fractional Laplacian. We give concrete examplemodels to illustrate these curious situations and prove Poincar\'e inequalitiesfor the fractional Laplacians of any order on domains that are bounded in onedirection. We establish Runge approximation results in these general settings,improve regularity assumptions also in the cases of bounded sets and provegeneral exterior determination results. Counterexamples to uniqueness in theinverse fractional conductivity problem with partial data are constructed inanother companion work.

Fractional Calder\'on problems and Poincar\'e inequalities on unbounded domains