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Bochner-Riesz means at the critical index: Weighted and sparse bounds

David Beltran, Joris Roos, Andreas Seeger

TL;DR

This work develops sharp endpoint estimates for Bochner-Riesz means at the critical index on weighted spaces and establishes sparse domination in a broad range of exponents. Central to the approach is a refined decomposition of Riesz multipliers that yields strong kernel localization and preserves Fourier support, enabling a synthesis of vector-valued endpoint bounds with Calderón–Zygmund techniques. By combining these endpoint sparse bounds with recent weighted–sparse transfer results, the authors extend Vargas’ weak-type result to $L^p(w)\to L^{p,\infty}(w)$ for $A_1$ weights with explicit $p$-ranges, and obtain (fully optimal) sparse bounds for the special index $\lambda_* = \frac{d-1}{2d+2}$. The work advances the Bochner–Riesz program by bridging endpoint technology, vector-valued Bochner–Riesz estimates, and sparse domination to obtain robust weak-type and weighted inequalities with potential implications for the Bochner–Riesz conjecture in higher dimensions.

Abstract

We consider Bochner-Riesz means on weighted $L^p$ spaces, at the critical index $λ(p)=d(\frac 1p-\frac 12)-\frac 12$. For every $A_1$-weight we obtain an extension of Vargas' weak type $(1,1)$ inequality in some range of $p>1$. To prove this result we establish new endpoint results for sparse domination. These are almost optimal in dimension $d=2$; partial results as well as conditional results are proved in higher dimensions. For the means of index $λ_*=\frac{d-1}{2d+2}$ we prove fully optimal sparse bounds.

Bochner-Riesz means at the critical index: Weighted and sparse bounds

TL;DR

This work develops sharp endpoint estimates for Bochner-Riesz means at the critical index on weighted spaces and establishes sparse domination in a broad range of exponents. Central to the approach is a refined decomposition of Riesz multipliers that yields strong kernel localization and preserves Fourier support, enabling a synthesis of vector-valued endpoint bounds with Calderón–Zygmund techniques. By combining these endpoint sparse bounds with recent weighted–sparse transfer results, the authors extend Vargas’ weak-type result to for weights with explicit -ranges, and obtain (fully optimal) sparse bounds for the special index . The work advances the Bochner–Riesz program by bridging endpoint technology, vector-valued Bochner–Riesz estimates, and sparse domination to obtain robust weak-type and weighted inequalities with potential implications for the Bochner–Riesz conjecture in higher dimensions.

Abstract

We consider Bochner-Riesz means on weighted spaces, at the critical index . For every -weight we obtain an extension of Vargas' weak type inequality in some range of . To prove this result we establish new endpoint results for sparse domination. These are almost optimal in dimension ; partial results as well as conditional results are proved in higher dimensions. For the means of index we prove fully optimal sparse bounds.
Paper Structure (25 sections, 29 theorems, 190 equations, 1 figure)

This paper contains 25 sections, 29 theorems, 190 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Weighted estimates
  3. Sparse bounds
  4. A refined version of Theorem \ref{['thm:blackboxvariant']} and Bochner-Riesz type bounds for vector-valued functions
  5. Auxiliary inequalities on vector-valued functions
  6. Instances in which the hypothesis holds
  7. Decompositions of Riesz means
  8. The main estimates
  9. Proof of Theorem \ref{['thm:Xi']}
  10. Reduction to a linear operator
  11. Proof of Proposition \ref{['prop:p=p0']}
  12. Proof of Proposition \ref{['prop:p=1']}
  13. The case $p=1$, $r=2$: proof of \ref{['eq:p=1-r=2']}
  14. The case $p=1$, $r> 2$: proof of \ref{['eq:p=1--r']}
  15. Sparse domination, Part I
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $a>0$. For every $w\in A_1$ there exists an exponent $p_1(w)>1$ such that the operators ${\mathcal{R}}_{a,t}^{\lambda(p)}$ are bounded from $L^p(w)$ to $L^{p,\infty}(w)$ for $1 \leq p < p_1(w)$, uniformly in $t>0$. Moreover, $\lim_{t\to \infty} \| {\mathcal{R}}^{\lambda(p)}_{a,t} f-f\|_{L^{p,\in

Figures (1)

  • Figure 1: Sparse bounds for Riesz means ${\mathcal{R}}^\lambda_{a,t}$ in ${\mathbb {R}}^2$ for any $0 < \lambda < 1/2$ on the left, and for the special case $\lambda=1/6$ on the right. The blue boundary segments correspond to the new content of Theorems \ref{['thm:new-sparse-bound-2D']} and \ref{['thm:STendpt']}, resp. Similar figures hold for $d \geq 3$ for a restricted range of $\lambda$; see Remarks after Theorem \ref{['thm:blackboxvariant']}, and Theorem \ref{['thm:STendpt']}.

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4: BRS-expository
  • Lemma 3.1
  • ...and 41 more