Sobolev improving for averages over curves in ${\mathbf{R}}^4$

Abstract

We study $L^p$-Sobolev improving for averaging operators $A_{\gamma }$ given by convolution with a compactly supported smooth density ${\mu }_{\gamma }$ on a non-degenerate curve. In particular, in 4 dimensions we show that $A_{\gamma }$ maps $L^p({\mathbb{R}}^4)$ the Sobolev space $L_{1/p}^p({\mathbb{R}}^4)$ for all $6<p<\infty$. This implies the complete optimal range of $L^p$-Sobolev estimates, except possibly for certain endpoint cases. The proof relies on decoupling inequalities for a family of cones which decompose the wave front set of ${\mu }_{\gamma }$. In higher dimensions, a new non-trivial necessary condition for $L^p({\mathbb{R}}^n)\to L_{1/p}^p({\mathbb{R}}^n)$ boundedness is obtained, which motivates a conjectural range of estimates.

60 pages, 4 figures. Revised version incorporating the referee's suggestions. To appear in Advances in Mathematics