On the Oberlin affine curvature condition

Abstract

In this paper we generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension $d$ in ${\mathbb{R}}^n$ , $1\le d\le n-1$. We show that a canonical affine invariant measure exists and that, modulo sufficient regularity assumptions on the submanifold, the measure satisfies the affine curvature condition of D. Oberlin with an exponent which is best possible. The proof combines aspects of Geometric Invariant Theory, convex geometry, and frame theory. A significant new element of the proof is a generalization to higher dimensions of an earlier result of the author concerning inequalities of reverse Sobolev type for polynomials on arbitrary measurable subsets of the real line.

28 pages, 2 figures

References in corpus (3)

• Universal L^p improving for averages along polynomial curves in low dimensions
• Uniform estimates for Fourier restriction to polynomial curves in ${\mathbb{R}}^d$
• A uniform Fourier restriction theorem for surfaces in R^d

Cited by in corpus (5)

• Decoupling inequalities for quadratic forms
• Geometric averaging operators and nonconcentration inequalities
• A restricted $2$-plane transform related to Fourier Restriction for surfaces of codimension $2$
• Uniform Fourier restriction for convex curves
• An Algebraic Brascamp-Lieb Inequality