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Dimension-free estimates on $l^2 (\mathbb{Z} ^d)$ for discrete dyadic maximal function over $l^1$ balls: small scales

Jakub Niksiński

Abstract

We give a dimension-free bound on $l^p(\mathbb{Z} ^d)$ for discrete Hardy-Littlewood operator over $l^1$ balls in $\mathbb{Z} ^d$ with small dyadic radii, where $p \in [2, \infty]$.

Dimension-free estimates on $l^2 (\mathbb{Z} ^d)$ for discrete dyadic maximal function over $l^1$ balls: small scales

Abstract

We give a dimension-free bound on for discrete Hardy-Littlewood operator over balls in with small dyadic radii, where .
Paper Structure (7 sections, 12 theorems, 128 equations)

This paper contains 7 sections, 12 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Estimate for $|B_t \cap \mathbb{Z}^d|$ and its consequences.
  4. Estimates for the multiplier $s_t$.
  5. Krawtchouk polynomials
  6. Introduction of new multipliers
  7. Conclusions and proof of the main theorem

Key Result

Theorem 1.1

For any $p \in [2,\infty)$, $d \in \mathbb{N}$, $d \geq 4$ and any $f \in l^p(\mathbb{Z} ^d)$ we have where $\mathbb{D}=\lbrace 2^n : n \in \mathbb{N}_0 \rbrace$ is the set of dyadic integers.

Theorems & Definitions (26)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • ...and 16 more