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We prove uniqueness and stability for the inverse boundary value problem ofthe two dimensional Schr\"odinger equation. We do not assume the potentials tobe continuous or even bounded. Instead, we assume that some of their positivefractional derivatives are in a specific Lorentz space. These spaces are anatural generalization to the usual fractional Sobolev spaces. The thesisconsists of two parts.  In the first part, we define the generalized fractional Sobolev spaces andprove some of their properties including embeddings and interpolationidentities. In particular we sharpen the usual Sobolev embedding into the spaceof H\"older-continuous functions, by showing that a particular kind of spaceembeds into the space of continuous functions without any modulus ofcontinuity.  The inverse problem is considered in the second part of the thesis. We provea new Carleman estimate for $\partial$. This estimate has a fast decay rate,which will allow us to consider potentials with very low regularity. After thatwe use Bukhgeim's oscillating exponential solutions, Alessandrini's identityand stationary phase to get information about the difference of the potentialsfrom the difference of the Cauchy data. The stability estimate will be oflogarithmic type, but works with potentials of low regularity.

On the Gel'fand-Calder\'on inverse problem in two dimensions