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Boundedness of some convolution-type operators on metric measure spaces

J. M. Aldaz

Abstract

We explore boundedness properties in the context of metric measure spaces, of some natural operators of convolution type whose study is suggested by certain transformations used in computer vision.

Boundedness of some convolution-type operators on metric measure spaces

Abstract

We explore boundedness properties in the context of metric measure spaces, of some natural operators of convolution type whose study is suggested by certain transformations used in computer vision.
Paper Structure (3 sections, 5 theorems, 25 equations)

This paper contains 3 sections, 5 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Additional definitions, and results
  3. Some remainders on the notion of $\tau$-additivity

Key Result

Theorem 2.5

Let $(X, d)$ be a metric space. If the space $(X, d)$ has the equal radius Besicovitch intersection property with constant $E$, then for every $r > 0$ and every $\tau$-additive, locally finite Borel measure $\mu$ on $X$, whenever $1\le p < \infty$ we have $\|A_{r, \mu}\|_{L^p \to L^p} \le E^{1/p}$

Theorems & Definitions (20)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Example 2.6
  • ...and 10 more