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$L^p$ improving properties and maximal estimates for certain multilinear averaging operators

Chu-hee Cho, Jin Bong Lee, Kalachand Shuin

Abstract

In this article we focus on $L^{p}$ estimates for two types of multilinear lacunary maximal averages over hypersurfaces with curvature conditions. Moreover, we give a different proof for the bilinear lacunary spherical maximal functions. To obtain our results, we make use of the $L^1$-improving estimates of multilinear averaging operators. We also obtain $L^p$-improving estimates for certain multilinear averages by means of the nonlinear Brascamp-Lieb inequality.

$L^p$ improving properties and maximal estimates for certain multilinear averaging operators

Abstract

In this article we focus on estimates for two types of multilinear lacunary maximal averages over hypersurfaces with curvature conditions. Moreover, we give a different proof for the bilinear lacunary spherical maximal functions. To obtain our results, we make use of the -improving estimates of multilinear averaging operators. We also obtain -improving estimates for certain multilinear averages by means of the nonlinear Brascamp-Lieb inequality.
Paper Structure (21 sections, 21 theorems, 159 equations)

This paper contains 21 sections, 21 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Proofs of Propositions \ref{['prop-qbanach esti']} and \ref{['prop_qbesti_cM']}
  3. Proof of Proposition \ref{['prop-qbanach esti']}
  4. Proof of Proposition \ref{['prop_qbesti_cM']}
  5. $L^1$- improving estimates \ref{['ineq_L1_impr']}
  6. Quasi-Banach space estimates \ref{['ineq_qba_cM']}
  7. smoothing estimates \ref{['ineq_reg_cM']}
  8. A Nonlinear Brascamp-Lieb inequality approach to $L^p$-improving estimates for ${\mathcal{A}}_{\mathcal{S}}^\Theta$
  9. Nonlinear Brascamp-Lieb inequality
  10. Proof of Theorem \ref{['thm-lp improving']}
  11. $d>d_K > k$
  12. $k \geq d_K\geq1$
  13. Proof of Theroem \ref{['thm-lac']}
  14. Proofs of Lemmas \ref{['lem-qba esti']} and \ref{['lem-smoothing']}
  15. Proof of Lemma \ref{['lem-qba esti']}
  16. ...and 6 more sections

Key Result

Proposition 1.1

Let $\mathcal{A}_{\mathcal{S}}^\Theta(\mathrm{F})$ be given in defn-mlao and $\mathcal{S}$ be a compact smooth hypersurface contained in $\mathbb{B}^{d}(0,1)$ with $\kappa\leq d-1$ nonvanishing principal curvatures. Let $\Theta = \{\Theta_j\}_{j=1}^m$ be a family of mutually linearly independent rot whenever $1\leq \frac{1}{p} \leq \frac{2(\kappa+1)}{\kappa+2}=\sum_{j=1}^m\frac{1}{p_j}$.

Theorems & Definitions (30)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 20 more