On curvature and the bilinear multiplier problem

TL;DR

This paper establishes that certain curvature conditions on the boundary of a domain in four-dimensional space ensure the unboundedness of a specific bilinear Fourier multiplier operator outside the local L^2 setting, especially for locally strictly convex domains.

Contribution

It introduces curvature-based criteria guaranteeing unboundedness of bilinear Fourier multipliers for domains with locally strictly convex boundaries.

Findings

Curvature conditions imply unboundedness of the bilinear multiplier.

Locally strictly convex boundaries satisfy these curvature conditions.

Results extend the understanding of bilinear multiplier behavior beyond L^2.

Abstract

We provide sufficient normal curvature conditions on the boundary of a domain $D \subset \BBR^4$ to guarantee unboundedness of the bilinear Fourier multiplier operator $\T_D$ with symbol $\chi_D$ outside the local $L^2$ setting, \textit{i.e}. from $L^{p_1} (\BBR^2) \times L^{p_2} (\BBR^2) \to L^{p_3'} (\BBR^2)$ with $\sum \frac{1}{p_j} = 1$ and $p_j <2$ for some $j$. In particular, these curvature conditions are satisfied by any domain $D$ that is locally strictly convex at a single boundary point.