Bilinear identities involving the $k$-plane transform and Fourier extension operators

Abstract

We prove certain $L^2({\mathbb{R}}^n)$ bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the $k$-plane transform. As the estimates are $L^2$-based, they follow from bilinear identities: in particular, these are the analogues of a known identity for paraboloids, and may be seen as higher-dimensional versions of the classical $L^2({\mathbb{R}}^2)$-bilinear identity for Fourier extension operators associated to curves in ${\mathbb{R}}^2$.

arXiv comment21 pages, 4 figures. Revised version incorporating referee suggestions. To appear in Proc. A Royal Soc. Edinburgh

References in corpus (6)

• Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid
• Some sharp restriction inequalities on the sphere
• Extremizers for Fourier restriction on hyperboloids
• A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions
• On sharp bilinear Strichartz estimates of Ozawa-Tsutsumi type
• A sharp $k$-plane Strichartz inequality for the Schrödinger equation

Cited by in corpus (1)

• Tomography bounds for the Fourier extension operator and applications