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Improved curvature conditions on $L^2\times\cdots\times L^2 \to L^{2/m}$ bounds for multilinear maximal averages

Chuhee Cho, Jin Bong Lee, Kalachand Shuin

Abstract

In this article, we focus on $L^{2}(\mathbb{R}^d)\times\cdots\times L^{2}(\mathbb{R}^d)\rightarrow L^{2/m}(\mathbb{R}^d)$ estimates for multilinear maximal averages over non-degenerate hypersurfaces. Our findings is new for $m$-linear averages with $m\geq3$, and represent a reproof of the recent result of T. Borges, B. Foster, and Y. Ou on the curvature conditions of the hypersurfaces required in establishing $L^{2}(\mathbb{R}^d)\times L^{2}(\mathbb{R}^d)\rightarrow L^{1}(\mathbb{R}^d)$ estimates of bilinear maximal functions.

Improved curvature conditions on $L^2\times\cdots\times L^2 \to L^{2/m}$ bounds for multilinear maximal averages

Abstract

In this article, we focus on estimates for multilinear maximal averages over non-degenerate hypersurfaces. Our findings is new for -linear averages with , and represent a reproof of the recent result of T. Borges, B. Foster, and Y. Ou on the curvature conditions of the hypersurfaces required in establishing estimates of bilinear maximal functions.
Paper Structure (8 sections, 12 theorems, 70 equations)

This paper contains 8 sections, 12 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\alpha$-dimensional measures and weighted estimates
  4. $L^p$ improving estimates for multilinear operators with the limited decay condition
  5. Proof of Theorem \ref{['thm_loc']}
  6. Proof of Lemma \ref{['lem_loc_max']}
  7. Proof of Theorem \ref{['thm_main']}
  8. Proof of Lemma \ref{['lem_decay1']}

Key Result

Theorem 1.1

Let $\sigma$ be the surface measure of a compact and smooth surface $\Sigma$ without boundary such that $k$ of its $2d-1$ principal curvatures are non-zero. Then $\mathrm{M}_\sigma$ maps $L^{2}(\mathbb{R}^{d})\times L^{2}(\mathbb{R}^{d})\rightarrow L^{1}(\mathbb{R}^{d})$ when $k>d+2$.

Theorems & Definitions (19)

  • Theorem 1.1: CGHHS_2022, Theorem 2
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • ...and 9 more