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Lebesgue bounds for multilinear spherical and lacunary maximal averages

Xinyu Gao

Abstract

We establish $L^{p_1}(\mathbb R^d) \times \cdots \times L^{p_n}(\mathbb R^d) \rightarrow L^r(\mathbb R^d)$ bounds for spherical averaging operators $\mathcal A^n$ in dimensions $d \geq 2$ for indices $1\le p_1,\dots , p_n\le \infty$ and $\frac{1}{p_1}+\cdots +\frac{1}{p_n}=\frac{1}{r}$. We obtain this result by first showing that $\mathcal A^n$ maps $L^1 \times \cdots \times L^1 \rightarrow L^1$. We also obtain similar estimates for lacunary maximal spherical averages in the largest possible open region of indices.

Lebesgue bounds for multilinear spherical and lacunary maximal averages

Abstract

We establish bounds for spherical averaging operators in dimensions for indices and . We obtain this result by first showing that maps . We also obtain similar estimates for lacunary maximal spherical averages in the largest possible open region of indices.

Paper Structure

This paper contains 7 sections, 11 theorems, 116 equations.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Bounds for the spherical average operator Aⁿ
  4. Case $d=2$.
  5. Case $d>2$.
  6. Bounds for the lacunary spherical maximal operator $\mathcal{M}_{\mathrm{lac}}^{n}$
  7. Visualization

Key Result

Theorem 1

Assume $d\ge 2$ and $1\le p_1,\dots,p_n\le\infty$ with $\tfrac{1}{r}=\tfrac{1}{p_1}+\cdots+\tfrac{1}{p_n}$. Then the $n$-linear spherical average operator $\mathcal{A}^n$ (defined in AOP) maps $L^{p_1}(\mathbb R^d)\times\cdots\times L^{p_n}(\mathbb R^d)\to L^r(\mathbb R^d)$.

Theorems & Definitions (25)

  • Theorem 1
  • Lemma 1.1
  • Definition 1
  • Definition 2
  • Lemma 1.2: Slicing Formula
  • proof
  • Lemma 1.3: Change of Variables
  • proof
  • Definition 3
  • Proposition 1.1
  • ...and 15 more