Fourier transform restriction and fractal sets of R

Fourier transform can be extended to a surjective isometry on L^2 spaces, but doing so implies losing continuity and the ability to study the transform of your function on sets of measure zero. By interpolation one can define a Fourier transform for functions in L^p for 1<p<2, whose transforms are not continuous, but they are well defined on some sets of measure 0 (i.e. hypersurfaces) in the sense of L^2 functions, for some suitable choice of p. In the thesis we look at the study that have been conducted over this subject. In dimension d>2, a required property for a hypersurface over which we would like to establish a restriction of Fourier transform is non-vanishing curvature in each point of the set. However, more recently, it has been discovered that a similar theory can be formulated even in dimension d=1 where the notion of curvature has no meaning. In fact, we will see how similar properties can be derived through the concept of Fourier dimension, which can be seen in a way as an extension of the concept of curvature to dimension 1. We will also construct sets in dimension 1 for which a restriction of Fourier transform is possible for L^p functions with suitable values of p>1 (Salem sets).