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Sparse bounds for maximal triangle and bilinear spherical averaging operators

Eyvindur Ari Palsson, Sean R. Sovine

TL;DR

The paper extends sparse domination methods to a general bilinear convolution framework L_t with compactly supported measures, showing that interior L^p-improving and continuity estimates for L_t and its unit-scale maximal variant yield sparse bounds for lacunary and full maximal operators associated with bilinear spherical and triangle averages. By developing an abstract sparse domination theorem and verifying the necessary continuity bounds for these operators (via multipliers, decay, and dimension-dependent results), the authors establish sharp sparse bounds in interior exponent regions and derive weighted bounds through extrapolation. The work unifies and extends prior bilinear sparse results (e.g., RSS) to non-product-type operators and to a broader class of measures, with explicit results for lacunary and full maximal operators and implications for weighted inequalities. The analysis relies on dyadic stopping-time decompositions, Calderón–Zygmund-type decompositions, and carefully crafted continuity estimates for uniformly decaying bilinear multipliers, applied to bilinear spherical and triangle averaging operators and their maximal variants.

Abstract

We show that the method in recent work of Roncal, Shrivastava, and Shuin can be adapted to show that certain $L^p$-improving bounds in the interior of the boundedness region for the bilinear spherical or triangle averaging operator imply sparse bounds for the corresponding lacunary maximal operator, and that $L^p$-improving bounds in the interior of the boundedness region for the corresponding single-scale maximal operators imply sparse bounds for the correpsonding full maximal operators. More generally we show that the framework applies for bilinear convolutions with compactly supported finite Borel measures that satisfy appropriate $L^p$-improving and continuity estimates. This shows that the method used by Roncal, Shrivastava, and Shuin can be adapted to obtain sparse bounds for a general class of bilinear operators that are not of product type, for a certain range of $L^p$ exponents.

Sparse bounds for maximal triangle and bilinear spherical averaging operators

TL;DR

The paper extends sparse domination methods to a general bilinear convolution framework L_t with compactly supported measures, showing that interior L^p-improving and continuity estimates for L_t and its unit-scale maximal variant yield sparse bounds for lacunary and full maximal operators associated with bilinear spherical and triangle averages. By developing an abstract sparse domination theorem and verifying the necessary continuity bounds for these operators (via multipliers, decay, and dimension-dependent results), the authors establish sharp sparse bounds in interior exponent regions and derive weighted bounds through extrapolation. The work unifies and extends prior bilinear sparse results (e.g., RSS) to non-product-type operators and to a broader class of measures, with explicit results for lacunary and full maximal operators and implications for weighted inequalities. The analysis relies on dyadic stopping-time decompositions, Calderón–Zygmund-type decompositions, and carefully crafted continuity estimates for uniformly decaying bilinear multipliers, applied to bilinear spherical and triangle averaging operators and their maximal variants.

Abstract

We show that the method in recent work of Roncal, Shrivastava, and Shuin can be adapted to show that certain -improving bounds in the interior of the boundedness region for the bilinear spherical or triangle averaging operator imply sparse bounds for the corresponding lacunary maximal operator, and that -improving bounds in the interior of the boundedness region for the corresponding single-scale maximal operators imply sparse bounds for the correpsonding full maximal operators. More generally we show that the framework applies for bilinear convolutions with compactly supported finite Borel measures that satisfy appropriate -improving and continuity estimates. This shows that the method used by Roncal, Shrivastava, and Shuin can be adapted to obtain sparse bounds for a general class of bilinear operators that are not of product type, for a certain range of exponents.

Paper Structure

This paper contains 15 sections, 23 theorems, 149 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Specific Bounds for $T_t$, $B_t$, and their Maximal Variants
  3. Abstract Sparse Domination Theorem
  4. Proof of Lemma \ref{['lem1']} using Lemma \ref{['lem2']}.
  5. Proof of Theorem \ref{['thm1']} using Lemma \ref{['lem1']}.
  6. Proof of Lemma \ref{['lem2']}
  7. Continuity Estimates
  8. Uniformly Decaying Bilinear Multipliers
  9. Localized Maximal Operators for Uniformly Decaying Multipliers
  10. Bilinear Spherical Averaging Operators
  11. Triangle Averaging Operators
  12. Application to Triangle and Bilinear Spherical Averages
  13. Sparse Bounds for Lacunary Operators
  14. Sparse Bounds for $T_{\text{\normalfont full}}$, $B_{\text{\normalfont full}}$, and $L_{\text{\normalfont full}}$
  15. Weighted Bounds

Key Result

Theorem 1.1

Let $(1/p, 1/q, 1/r)$ with $r \geq p,q$ and $r > 1$ be in the interior of the boundedness set $\mathfrak{S}^d$, $d\geq 2$ of $B_{t}$. Then for any compactly supported bounded functions $f$, $g$ and $h$, where the supremum is over all sparse collections. Let $(1/p, 1/q, 1/r)$ with $r \geq p,q$ and $r > 1$ be in the interior of the boundedness set $\mathfrak{S}_{\star}^d$, $d\geq 4$, of $B_{\star

Figures (1)

  • Figure :

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: IPS, Thm. 1.2
  • Theorem 2.2: IPS, Thm. 1.1
  • Corollary 2.3
  • Theorem 2.4: CLM, Thm. 4
  • Corollary 2.5
  • Example 2.6
  • Theorem 2.7: JeongLee, Thm. 3.1
  • ...and 16 more