Outline: Throughout mathematics, the Fourier transform turns out to be an invaluable device. This project is devoted to exploring one of the earliest topics of study for the Fourier transform, namely its properties on Lp spaces. This theory is a cornerstone of harmonic analysis and can be immensely useful when analyzing PDEs.
The project will first deal with L1 functions, developing the basic properties of the Fourier transform including the Riemann–Lebesgue lemma. Then L2 functions will be considered, and the Fourier inversion theorem and the Plancherel theorem will be proved. Through the Marcinkiewicz interpolation theorem, the Fourier transform will be extended to Lp spaces in between L1 and L2, and if there is time then some results on Lp spaces for p>2 will be explored as well.
Sources: - Classical and Multilinear Harmonic Analysis, Camil Muscalu,