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 $$ \Big| \iint_{\mathbb{R}^d\times\mathbb{R}^d} f(x)\,g(y)\,e^{i\lambda\phi(x,y)}\,\eta(x,y)\,dx\,dy\Big|\le C|\lambda|^{-\gamma} \|{f}\|_{L^2}\|{g}\|_{L^2}$$ where $\lambda\in\mathbb{R}$ is a large parameter, $\phi$ 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 $C<\infty$ depend on $\phi$ 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 $\phi$ and the product structure of $f(x)\,g(y)$. If $\phi$ is nonconstant then $e^{i\lambda\phi}$ oscillates rapidly, creating cancellation that potentially results in smallness of the integral.
Implicitly oscillatory integrals involve no overtly oscillatory factor $e^{i\lambda\phi}$; 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 \[ \int_{\mathbb{R}^2} \prod_{j=1}^N (f_j\circ\varphi_j)(x)\,\eta(x)\,dx\] where $\varphi_j:\mathbb{R}^2\to\mathbb{R}^1$ 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.