Restrição de Fourier em superfícies de Rn e estimativas lineares e bilineares de Strichartz - Santos, Ian Rodrigues dos (2020)
In this work we study oscillatory integrals of the first type and via the stationary method we show an optimal decay of the Fourier transform of measures whose support belongs to smooth surfaces with m non-zero principal curvatures. We use this decay to show Fourier constraint theorems on compact surfaces and quadratic surfaces. Using these ideas, we show Strichartz linear estimates in Lebesgue space Lp for the wave and Schrödinger equations. We finish the dissertation with Strichartz’s bilinear estimates to show the end-point case of the permissible sharp point P = (2, 2σ/σ−1) for a dispersive evolution group {U(t)}t∈R satisfying energy and dispersion estimates. Consequently, we prove the end-point case for Schrödinger’s homogeneous Strichartz estimates and for wave equation on high dimension, namely, n ≥ 4.