Revista Matemática Iberoamericana

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Volume 28, Issue 2, 2012, pp. 351–369
DOI: 10.4171/rmi/680

Published online: 2012-04-22

On curvature and the bilinear multiplier problem

S. Zubin Gautam[1]

(1) Indiana University, Bloomington, USA

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.

Keywords: Bilinear Fourier multipliers, multilinear operators

Gautam S. Zubin: On curvature and the bilinear multiplier problem. Rev. Mat. Iberoam. 28 (2012), 351-369. doi: 10.4171/rmi/680