On continuum incidence problems related to harmonic analysis

Abstract

We consider certain estimates involving averaging operators over curves and hypersurfaces that can be cast into a combinatorial framework. We show that hypersurfaces with nonzero rotational curvature satisfy the usual restricted weak-type bound, but our proof does not involve the Fourier transform. Secondly, we show that a Strichartz-type estimate for the wave equation in 2+1 dimensions can be obtained in a similar fashion, and we give a simplified proof of Wolff's endpoint theorem for maximal averages over circles. Finally, examples are provided that show what the optimal bound can be for the tangency problem of circles in the plane.