Michael Christ (UC Berkeley)

The APDE seminar on Monday, 10/18, will be given by our own Michael Christ online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: On quadrilinear implicitly oscillatory integrals

Abstract: Multilinear oscillatory integrals arise
in various contexts in harmonic analysis,
in partial differential equations, in ergodic theory,
and in additive combinatorics.  We discuss the majorization of integrals
$\int \prod_{j} (f_j\circ\varphi_j)$ of finite products
by negative order norms of the factors,
where integration is over a ball in Euclidean space and $\varphi_j$ are smooth
mappings to a space of strictly lower dimension. The talk focuses
on the quadrilinear case, after work on the trilinear case of Bourgain (1988),
of Joly, M\’etivier, and Rauch (1995), and of the speaker (2019).
Sublevel set inequalities, which quantify the nonsolvability of certain systems
of linear equations, are a central element of the analysis.