Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Compactness of multilinear commutators generated by VMO functions and fractional integral operator on Morrey spaces

Daiki Takesako

TL;DR

This work investigates the compactness of multilinear commutators $[a_{\{1,...,l\}},I_{\alpha}]$ on Morrey spaces when the symbol $a_1$ belongs to ${\rm VMO}$. The authors develop a framework based on decompositions of tilde/ bar / star closed subspaces of Morrey spaces and leverage Adams-type boundedness to prove that $[a_{\{1,...,l\}},I_{\alpha}]$ is compact from ${\mathcal{M}}^{p}_{q}$ to $\widetilde{\mathcal{M}}^{s}_{t}$ under the scaling relations $\tfrac{1}{s}=\tfrac{1}{p}-\tfrac{\alpha}{n}>0$ and $\tfrac{p}{q}=\tfrac{s}{t}$. The proofs combine dyadic decomposition, $BMO$-$VMO$-driven tail estimates, and a density-approximation argument exploiting smooth compactly supported symbols to achieve compactness. These results extend the compactness theory for commutators on Morrey-type spaces to a broad multilinear setting and clarify approximation properties of such operators in non-homogeneous Morrey contexts.

Abstract

The aim of this paper is to improve the compactness of the multilinear commutators in Morrey spaces generated by VMO functions and fractional integral operators. In this paper, we will use the decomposition of the tilde closed subspaces of Morrey spaces. This gives us more understanding about commutators.

Compactness of multilinear commutators generated by VMO functions and fractional integral operator on Morrey spaces

TL;DR

This work investigates the compactness of multilinear commutators on Morrey spaces when the symbol belongs to . The authors develop a framework based on decompositions of tilde/ bar / star closed subspaces of Morrey spaces and leverage Adams-type boundedness to prove that is compact from to under the scaling relations and . The proofs combine dyadic decomposition, --driven tail estimates, and a density-approximation argument exploiting smooth compactly supported symbols to achieve compactness. These results extend the compactness theory for commutators on Morrey-type spaces to a broad multilinear setting and clarify approximation properties of such operators in non-homogeneous Morrey contexts.

Abstract

The aim of this paper is to improve the compactness of the multilinear commutators in Morrey spaces generated by VMO functions and fractional integral operators. In this paper, we will use the decomposition of the tilde closed subspaces of Morrey spaces. This gives us more understanding about commutators.
Paper Structure (8 sections, 10 theorems, 60 equations)

This paper contains 8 sections, 10 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterization and the decomposition of closed subspace of Morrey spaces
  4. Properties of operators
  5. Lemmas for the proof
  6. Proof of Theorem \ref{['main']}
  7. Proof of $[a_{\{1, ... , l\}},I_{\alpha}]f \in \overset{*}{{\mathcal{M}}}{}^s_t({\mathbb R}^n)$
  8. Proof of $[a_{\{1, ... , l\}},I_{\alpha}]f \in \overline{{\mathcal{M}}}{}^s_t({\mathbb R}^n)$

Key Result

Theorem 1.6

Suppose $1 < q \le p < \infty$, $l \in \mathbb{N}$, and $a_1 \in {\rm VMO}({\mathbb R}^n)$. Take $s, t$ so that Then the operator $[a_{\{1, ... ,l\}},I_{\alpha}]$ is compact from ${\mathcal{M}}^{p}_q({\mathbb R}^n)$ to $\widetilde{\mathcal{M}}^{s}_t({\mathbb R}^n)$.

Theorems & Definitions (18)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • ...and 8 more