Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The distance in Morrey spaces to $C^{\infty}_{\mathrm{comp}}$

Satoshi Yamaguchi

Abstract

In this paper we characterize the distance between the function $f$ and the set $C^{\infty}_{\mathrm{comp}}(\mathbb{R}^d)$ in generalized Morrey spaces $L_{p,φ}(\mathbb{R}^d)$ with variable growth condition. We also prove that the bi-dual of $\overline{C^{\infty}_{\mathrm{comp}}(\mathbb{R}^d)}^{L_{p,φ}(\mathbb{R}^d)}$ is $L_{p,φ}(\mathbb{R}^d)$. As an application of the characterization of the distance we show the boundedness of Calderón-Zygmund operators on $\overline{C^{\infty}_{\mathrm{comp}}(\mathbb{R}^d)}^{L_{p,φ}(\mathbb{R}^d)}$. By the duality we also see that these operators are bounded on its dual and bi-dual spaces.

The distance in Morrey spaces to $C^{\infty}_{\mathrm{comp}}$

Abstract

In this paper we characterize the distance between the function and the set in generalized Morrey spaces with variable growth condition. We also prove that the bi-dual of is . As an application of the characterization of the distance we show the boundedness of Calderón-Zygmund operators on . By the duality we also see that these operators are bounded on its dual and bi-dual spaces.

Paper Structure

This paper contains 7 sections, 23 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Mollifier
  3. Proof of Theorem \ref{['thm:distance']}
  4. Proof of $d(f,C^{\infty}_{\mathrm{comp}}(\mathbb{R}^d))\le CA_{p,\phi}(f)$
  5. Proof of $A_{p,\phi} \le d(f,C^{\infty}_{\mathrm{comp}}(\mathbb{R}^d))$
  6. Proof of Theorem \ref{['thm:bidual']}
  7. Calderón--Zygmund operators

Key Result

Theorem 1.1

Let $p\in[1,\infty)$, and let $\phi$ be in $\mathcal{G}^{\rm dec}_p$ and satisfy NC. Then there exists a positive constant $C$ such that, for all $f\in L_{p,\phi}(\mathbb{R}^d)$, Moreover, if then, for all $f\in L_{p,\phi}(\mathbb{R}^d)$,

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark 1.1
  • Corollary 1.2
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Lemma 2.2
  • ...and 34 more