Local smoothing and maximal estimates for average over surfaces of codimension 2 in $\mathbb R^4$

TL;DR

The paper develops sharp local smoothing and maximal estimates for averages over codimension-2 surfaces in $\mathbb{R}^4$ by reducing the averaging problem to Fourier extension operators on a conical extension $\Sigma$ in $\mathbb{R}^5$. It combines an induction-on-scale argument with $L^2$--$L^3$ trilinear restriction estimates and $L^4$ decoupling to establish sharp $L^p$--$L^q$ local smoothing for a broad class of nondegenerate surfaces, including a key model $\Gamma_\circ$. It further extends these ideas to sharp $L^p$--$L^q$ maximal estimates for maximal averages over half-dimension surfaces in even dimensions, using Greenleaf restriction theory and Bourgain interpolation. The results identify the precise ranges of exponents and provide a framework that unifies local smoothing and maximal estimates via conic extensions and multilinear analysis. The work advances understanding of local smoothing for intermediate codimension submanifolds and demonstrates the power of decoupling and multilinear restriction techniques in higher codimension settings.

Abstract

In this paper, we obtain local smoothing estimates for the averages over nondegenerate surfaces of codimension $2$ in $\mathbb R^4$. We make use of multilinear restriction estimates and decoupling inequalities for a hypersurface in $\mathbb R^5$, a conical extension of a two-dimensional nondegenerate surface along two flat directions. We also establish sharp $L^p$--$L^q$ estimates for maximal averages over nondegenerate surfaces of half the ambient dimension in $\mathbb R^{2n}$ for even $n \ge 2$.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Corollary 1.5
• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• Lemma 2.4