The almost optimal multilinear restriction estimate for hypersurfaces with curvature: the case of $n-1$ hypersurfaces in R^n

Abstract

In this paper we establish the optimal multilinear restriction estimate for n − 1 hypersurfaces with some curvature, where n is the dimension of the underlying space. The result is sharp up to the endpoint and the role of curvature is made precise in terms of the shape operator.

A note on bilinear wave-Schrödinger interactions

“…It is conjectured that the R ǫ loss can be removed, but this is currently an open question (however see [1] and [2] for recent progress). The U 2 version of Theorem 6 is then the following.…”

“…Financial support by the Marsden Fund Council grant 19-UOO-142, and the German Research Foundation (DFG) through the CRC 1283 "Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications" is acknowledged. 1 Here ∡(x, y) = (1…”

$L^p-L^q$ local smoothing estimates for the wave equation via $k$-broad Fourier restriction

“…valid for any 1 < p ≤ q < ∞ and any t > 0, and where the implicit constant is locally bounded in t. Here p = p/(p − 1). Similarly, any local smoothing estimate (1.2) can be interpolated with (1.5) to obtain L p (R n ) − L q (R n × [1,2]) estimates for q ≥ p. This motivates the following conjecture [31,38].…”

“…denotes a ball of radius R and the estimates are supposed to hold for all ε > 0 and all R ≥ 1. The only known cases are k = 2 [40,37], k = n [1] and k = n + 1 [3]. The remaining cases 3 ≤ k < n are open up to some partial positive results for p ≥ 2k k−1 [3].…”