This research project in Harmonic Analysis is concerned with boundedness properties of oscillatory integral operators in Lebesgue spaces as well as analogous problems in a discrete setting, i.e. exponential sum operators corresponding to classical Gaussian sums over finite fields. Bounds on these types of operators are fundamental in understanding summation of n-dimensional Fourier series, Fourier multipliers, regularity properties of solutions of partial differential equation and problems in integral geometry. Related to these problems are questions concerning compression phenomena of families of linear subspaces (Kakeya sets) and associated maximal function inequalities. The discrete case serves here as a model to develop combinatorial tools to get a better understanding in the Euclidean setting.
The proposed project focuses on questions arising from the problem of approximating a given signal by a superposition of pure tones.Depending on the approximation method employed one is lead to analyse various oscillatory integral operators. These operators arise also as solutions of fundamental equations in mathematical physics such as e.g. the wave equation and the KdV equation. Regularity and existence problems of this equations are intimately related to sharp bounds on the corresponding oscillatory integral operators.