The Triangle Operator

Abstract

We examine the averaging operator $T$ corresponding to the manifold in ${\mathbb{R}}^{2d}$ of pairs of points $(u,v)$ satisfying $|u|=|v|=|u-v|=1$, so that $\{0,u,v\}$ is the set of vertices of an equilateral triangle. We establish $L^p\times L^q\to L^r$ boundedness for $T$ for $(1/p,1/q,1/r)$ in the convex hull of the set of points $\{(0,0,0),(1,0,1),(0,1,1),(1/p_d,1/p_d,2/p_d)\}$, where $p_d=\frac{19d-4}{11d-12}$ and $d\ge 7$.

15 pages, discussion on the maximal variant expanded

References in corpus (1)

• Maximal estimates for the bilinear spherical averages and the bilinear Bochner-Riesz operators

Cited by in corpus (2)

• Multilinear maximal operators associated to simplices
• The pyramid operator