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.
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