Research Projects

In this project, we study boundedness of Fourier integral operators in different setting. First, our aim is to find the suitable conditions on symbols such that the corresponding multilinear Fourier integral operator on the product of modulation space is bounded. A key point to prove the above result is that we decompose the operator $T$ into its degenerate and non-degenerate components, denoted as $T_{\text {deg }}$ and $T_{\text {non-deg. }}$. This is accomplished through the concept of curvature, which quantifies the degree to which the phase function deviates from linearity. Next, our target is to obtain the boundedness of multilinear Fourier integral operators on the product of modulation space, and some dispersive estimates for parametrized families of periodic Fourier integral operators. To get the desired results, first we will deduce the result for trigonometric polynomials using the dominated convergence theorem for continuous periodic function. Then, using denseness, we will show that this restriction admits a unique bounded multilinear extension on the product of modulation space. In the end, our aim is to demonstrate the boundedness for multilinear Fourier integral operators with rough amplitudes and phase function that satisfies certain decay properties. This can be proved by estimating the low-frequency and high-frequency components of the integral separately. To demonstrate the boundedness of the low-frequency component, essential tools include Schur's lemma and the mean value theorem. For high frequency part, we will use the well-known Littlewood–Paley decomposition for FIOs.