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

TL;DR

This paper establishes optimal Sobolev improving estimates for averaging operators over non-degenerate curves in four-dimensional space, using decoupling inequalities and analyzing higher-dimensional cases.

Contribution

It proves the full range of $L^p$-Sobolev improving estimates for these operators in 4D and introduces new necessary conditions in higher dimensions.

Findings

Averaging operators map $L^p$ to $L^p_{1/p}$ for $6 < p < \infty$ in 4D.

Decoupling inequalities are key to the proof.

New necessary conditions for boundedness in higher dimensions.

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 $L^p_{1/p}(\mathbb{R}^4)$ 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 $L^p(\mathbb{R}^n) \to L^p_{1/p}(\mathbb{R}^n)$ boundedness is obtained, which motivates a conjectural range of estimates.