| We consider the problem of Lp- L2 restriction of the Fourier transform to quadratic submanifolds, that is, submanifolds of Euclidean space which are locally the graphs of homogeneous quadratic polynomials. The cases of codimension 1 and 2 have been understood for some time, whereas little is known for codimension greater than 3. In this dissertation, we develop some new techniques for proving restriction theorems for quadratic submanifolds. These techniques utilize aspects of the theory of resolution of singularities as well as some new results on oscillatory integrals involving polynomials of one variable. Our methods allow us to prove restriction theorems in several cases where the codimension is greater than 3. We also investigate the case of codimension 3 in greater detail. |
