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This paper is devoted to the problem of recovering a potential $q$ in adomain in $\mathbb{R}^d$ for $d \geq 3$ from the Dirichlet to Neumann map. Thisproblem is related to the inverse Calder\'on conductivity problem via theLiouville transformation. It is known from the work of Haberman and Tataru [11]and Nachman and Lavine [17] that uniqueness holds for the class ofconductivities of one derivative and the class of $W^{2,d/2}$ conductivitiesrespectively. The proof of Haberman and Tataru is based on the construction ofcomplex geometrical optics (CGO) solutions initially suggested by Sylvester andUhlmann [22], in functional spaces introduced by Bourgain [2]. The proof of thesecond result, in the work of Ferreira et al. [10], is based on theconstruction of CGO solutions via Carleman estimates. The main goal of thepaper is to understand whether or not an approach which is based on theconstruction of CGO solutions in the spirit of Sylvester and Uhlmann andinvolves only standard Sobolev spaces can be used to obtain these results. Infact, we are able to obtain a new proof of uniqueness for the Calder\'onproblem for 1) a slightly different class as the one in [11], and for 2) theclass of $W^{2,d/2}$ conductivities. The proof of statement 1) is based on anew estimate for CGO solutions and some averaging estimates in the same spiritas in [11]. The proof of statement 2) is on the one hand based on a generalizedSobolev inequality due to Kenig et al. [14] and on another hand, only involvesstandard estimates for CGO solutions [22]. We are also able to prove theuniqueness of a potential for 3) the class of $W^{s, 3/s}$ ($\supsetneqq W^{2,3/2}$) conductivities with $3/2 < s < 2$ in three dimensions. As far as weknow, statement 3) is new.

Recovering a Potential from Cauchy Data via Complex Geometrical Optics Solutions