# On curvature and the bilinear multiplier problem

### Sachin Gautam

Indiana University, Bloomington, USA

## Abstract

We provide sufficient normal curvature conditions on the boundary of a domain $D \subset \mathbb{R}^4$ to guarantee unboundedness of the bilinear Fourier multiplier operator $\mathrm{T}_D$ with symbol $\chi_D$ outside the local $L^2$ setting, i.e., from $L^{p_1} ( \mathbb{R}^2) \times L^{p_2} ( \mathbb{R}^2) \rightarrow L^{p_3'} ( \mathbb{R}^2)$ 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.

## Cite this article

Sachin Gautam, On curvature and the bilinear multiplier problem. Rev. Mat. Iberoam. 28 (2012), no. 2, pp. 351–369

DOI 10.4171/RMI/680