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The Pointillist principle for variation operators and jump functions

Kevin Hughes

Abstract

We extend the pointillist principles of Moon and Carrillo--de Guzmán to variational operators and jump functions.

The Pointillist principle for variation operators and jump functions

Abstract

We extend the pointillist principles of Moon and Carrillo--de Guzmán to variational operators and jump functions.

Paper Structure

This paper contains 3 sections, 6 theorems, 42 equations.

Table of Contents

  1. The pointillist principle
  2. Moon's theorem for variations
  3. Carrillo--de Guzmán's Theorem for variations

Key Result

Theorem 1

Suppose that $(T_m)_{m \in \mathbb{N} }$ is a sequence of convolution operators given by $T_m f := f * g_m$ with $g_m \in L^1(\mathbb{R} ^d)$ for each $m\in\mathbb{N}$. For any $q,r \geq 1$, $V_r(T_m f : m \in \mathbb{N} )$ is restricted weak-type $(1,q)$ with norm $C$ if and only if $V_r(T_m f : m

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 2.1: Moon's pointillist principle
  • proof : Proof of Proposition \ref{['proposition:Moon_approximation']}
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['theorem:Moon_variations']}
  • proof : Proof of Theorem \ref{['theorem:Moon_smoothing']}
  • Proposition 3.1
  • ...and 2 more