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

TL;DR

This paper develops local smoothing and maximal estimate results for averages over codimension 2 surfaces in four-dimensional space, utilizing multilinear restriction and decoupling techniques, and extends sharp $L^p$--$L^q$ estimates to higher even dimensions.

Contribution

It introduces new local smoothing estimates for surface averages in $\,\mathbb R^4$ and establishes sharp maximal $L^p$--$L^q$ bounds in higher even dimensions, advancing harmonic analysis techniques.

Findings

Established local smoothing estimates for codimension 2 surfaces in $\,\mathbb R^4$.

Derived sharp $L^p$--$L^q$ estimates for maximal averages in $\,\mathbb R^{2n}$ for even $n\ge 2$.

Utilized multilinear restriction and decoupling inequalities for hypersurfaces in $\,\mathbb R^5$.

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 $n \ge 2$.