L^p regularity of averages over curves and bounds for associated maximal operators

TL;DR

This work studies $L^p$ regularity for averages along finite-type curves in ${\mathbb R}^3$ and the associated maximal operator, establishing $L^p\to L^p_{1/p}$ smoothing for large $p$ and $L^p$-boundedness of the maximal operator above a Wolff-exponent threshold. The authors develop variations of Wolff's cone-multiplier inequality for curved cones, implement a microlocal smoothing framework for curves in higher dimensions, and apply these results to obtain local smoothing and maximal bounds for curves in ${\mathbb R}^3$, including a two-parameter bound for helices. Central to the approach is a cone-decomposition/plate-structure analysis and an induction-on-scales argument that transfers light-cone estimates to general curved cones. These methods connect cone-multiplier theory with geometric properties of curves, yielding sharp regularity results in high codimension and informing Kakeya-type geometric questions via two-parameter maximal estimates.

Abstract

We prove that for a finite type curve in $\mathbb R^3$ the maximal operator generated by dilations is bounded on $L^p$ for sufficiently large $p$. We also show the endpoint $L^p \to L^{p}_{1/p}$ regularity result for the averaging operators for large $p$. The proofs make use of a deep result of Thomas Wolff about decompositions of cone multipliers.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Proposition 2.1
• proof
• Corollary 2.2
• Proposition 3.1
• Lemma 3.2
• proof