On $[-1,1]$ which of the functions $f(x)=10e^{-100\pi x^{2}}$ or $g(x)=\sin(100\pi x)$ is larger? The answer depends on how you measure size, $|f|$ achieves a higher peak value than $|g|$, but the average of $|g|$ is the larger of the two.

To capture these competing notions of size, we typically use families of norms, such as the $L^{p}$ norms, which interpolate between “height” and “spread.” A natural question then arises: if we know a supremum bound for a class of functions, at how many points can a function from that class achieve this supremum? The Gaussian $f$ spikes sharply at $x=0$ and decays rapidly elsewhere, while $g$ is periodic and achieves its supremum at many points (but the values of $|g|$ at those points are much smaller than what $f$ achieves at $x=0$). We say that a function $f$ saturates a supremum bound if it achieves the bound at at least one point and it simultaneously saturates the supremum bound at $x$ and $y$ if both $|f(x)|$ and $|f(y)|$ achieve the supremum. How can we constrain the number of simultaneously saturating points?  

A very simple observation is that if both $|f(x)|$ and $|f(y)|$ are large, then so is $|f(x)f(y)|$. In this talk I will ``pull the thread'' on this observation and arrive at a very general technique for controlling the number of large values that a function can take. As an application I will demonstrate how to use the technique to prove $L^{p}$ estimates for multilinear restriction operators.