Special Analysis Seminar

Xiaochun Li
University of California, Los Angeles

On multilinear oscillatory integrals

Let v_1,v_2,\cdots,v_{n+1} be vectors in \mathbb{R}^{k+1}. And let Q(\bf{x}) be a polynomial. We call Q(\bf{x}) degenerate with respect to v_1,v_2,\cdots,v_{n+1} if

Q(\bf{x}) = \sum_{j=1}^{n+1}P_j(\bf{x}\cdot v_j)

for some one-dimensional polynomial P_1,\cdots,P_{n+1}.

Consider the form

\Lambda_\lambda(f_1, f_2, \cdots, f_{n+1}) = \int_{\mathbb{R}^{k+1}}
e^{\lambda Q(\bf{x})}\varphi(\bf{x}) \prod_{j=1}^{n+1}f_j(\bf{x}\cdot v_j)d\bf{x} ,

where \varphi is a standard bump function.

When n\leq 2k and the vectors v_j are in general position (that is, any k+1 vectors in {v_1,v_2,\cdots,v_{n+1}} are linearly independent), then we have

\big|\Lambda_\lambda(f_1, \cdots, f_{n+1})\big| \leq C \lambda^{-\varepsilon} \prod_{j=1}^{n+1}\|f_j\|_2

for all non-degenerate polynomials Q and some \varepsilon >0. Furthermore, the bound is uniform over all compact collections of non-degenerate polynomial Q.

This is a joint work with M. Christ, T. Tao and C. Thiele.

Copyright © 2004 UBC Mathematics Department