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Capacity in Besov and Triebel-Lizorkin spaces with generalized smoothness

Nijjwal Karak, Debarati Mondal

Abstract

We prove a lower bound estimate for capacities in Hajlasz-Besov, Hajlasz-Triebel-Lizorkin and Hajlasz-Sobolev spaces with generalized smoothness defined on metric spaces in terms of Netrusov-Hausdorff content or Hausdorff content.

Capacity in Besov and Triebel-Lizorkin spaces with generalized smoothness

Abstract

We prove a lower bound estimate for capacities in Hajlasz-Besov, Hajlasz-Triebel-Lizorkin and Hajlasz-Sobolev spaces with generalized smoothness defined on metric spaces in terms of Netrusov-Hausdorff content or Hausdorff content.
Paper Structure (7 sections, 9 theorems, 80 equations)

This paper contains 7 sections, 9 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Smoothness weight sequences and functions
  4. Hajł asz-Sobolev, Hajł asz-Besov and Hajł asz-Triebel-Lizorkin spaces with generalized smoothness
  5. $\gamma$-median
  6. Hausdorff content
  7. Proofs

Key Result

Theorem 1.2

Let $\phi\in \mathcal{A}_0$, $0<p<\infty$, $0<q<\infty$ and $\omega$ be any given function of admissible growth such that, for any $L\in \mathbb{Z}_{+}$, Let $x_0\in X,$$0<R<\infty.$ Then there are constants $C>0$ and $c>0$ such that, for any compact set $E\subset B(x_0,R),$ where, for any $r\in (0,R],$$h_\omega(r)=[\phi(r)\omega(r)]^p.$

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 13 more