L^2 decay estimates for oscillatory integral operators in several variables with homogeneous polynomial phases

Oscillatory integral operators mapping $L^2(mathbb R^{n_1})$ to $L^2(mathbb R^{n_2})$ play an important role in many problems in harmonic analysis and partial differential equations. We will briefly discuss the applicability of these operators in various contexts and give an overview of the current literature. We also mention recent results (joint with Allan Greenleaf and Wan Tang) where, extending earlier work of Phong and Stein in the case $n_1 = n_2 = 1$, we obtain optimal decay rates for the $L^2$ operator norm of oscillatory integral operators in $2+2$ variables with generic phases. Some other higher dimensional situations are also addressed.