The HADES seminar on Tuesday, May 3rd will be at 3:30 pm in Room 740.

Speaker: Michael Christ

Abstract: An archetypal (bilinear) oscillatory integral inequality states that $$|{\iint }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}f(x)g(y)e^{i\lambda \varphi (x,y)}\eta (x,y)dxdy|\le C|\lambda {|}^{-\gamma }\Vert f{\Vert }_{L^2}\Vert g{\Vert }_{L^2}$$ where $\lambda \in \mathbb{R}$ is a large parameter, $\varphi$ is a smooth real-valued phase function which is nondegenerate in a suitable sense, $f,g$ are arbitrary $L^2$ functions, $\eta$ is a  smooth compactly supported cutoff function, and $\gamma >0$ and $C<\mathrm{\infty }$ depend on $\varphi$ but not on $f,g,\lambda$. Its main features are the decaying factor $|\lambda {|}^{-\gamma }$, the absence of any smoothness hypothesis on the measurable factors $f,g$, and the interplay between the structure of $\varphi$ and the product structure of $f(x)g(y)$. If $\varphi$ is nonconstant then $e^{i\lambda \varphi }$ oscillates rapidly, creating cancellation that potentially results in smallness of the integral.

Implicitly oscillatory integrals involve no overtly oscillatory factor $e^{i\lambda \varphi }$; instead, the measurable factors $f_j$ are themselves assumed to be oscillatory, but in a less structured way. A multilinear integral of this type takes the form $${\int }_{{\mathbb{R}}^2}\prod _{j=1}^N(f_j\circ {\phi }_j)(x)\eta (x)dx$$ where ${\phi }_j:{\mathbb{R}}^2\to {\mathbb{R}}^1$ are smooth submersions, and the functions $f_j$ are merely measurable. The desired upper bound is expressed in terms of strictly negative order Sobolev norms of these functions. Thus the functions $f_j$ are rapidly oscillatory in the sense that they consist mainly of high frequency components.

I will give an introduction to recent (and ongoing) work on this topic, and on associatedsublevel set inequalities.