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Weighted Bourgain-Morrey-Besov type and Triebel-Lizorkin type spaces associated with operators

Tengfei Bai, Pengfei Guo, Jingshi Xu

Abstract

Let $(X,μ)$ be a space of homogeneous type satisfying $μ(X) =\infty$, the doubling property and the reverse doubling condition. Let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel enjoys a Gaussian upper bound. We introduce the weighted homogeneous Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces associated with the operator $L$. We obtain their continuous characterizations in terms of Peetre maximal functions, noncompactly supported functional calculus, heat kernel. Atomic and molecular decompositions of weighted homogeneous Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces are also given. As an application, we obtain the boundedness of the fractional power of $L$, the spectral multiplier of $L$ on Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces.

Weighted Bourgain-Morrey-Besov type and Triebel-Lizorkin type spaces associated with operators

Abstract

Let be a space of homogeneous type satisfying , the doubling property and the reverse doubling condition. Let be a nonnegative self-adjoint operator on whose heat kernel enjoys a Gaussian upper bound. We introduce the weighted homogeneous Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces associated with the operator . We obtain their continuous characterizations in terms of Peetre maximal functions, noncompactly supported functional calculus, heat kernel. Atomic and molecular decompositions of weighted homogeneous Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces are also given. As an application, we obtain the boundedness of the fractional power of , the spectral multiplier of on Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces.

Paper Structure

This paper contains 17 sections, 40 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dyadic cubes
  4. Muckenhoupt weights
  5. The class of distributions
  6. Bourgain-Morrey Besov type and Triebel-Lizorkin type spaces associated with $L$
  7. Definitions
  8. Characterization via Peetre maximal function
  9. Characterizations via compact supports
  10. Continuous characterizations by Schwartz functions
  11. Characterizations for weighted Bourgain-Morrey Triebel-Lizorkin type spaces via Lusin functions and Littlewood-Paley functions
  12. Atomic decompositions
  13. Atomic decompositions for $\dot{\mathcal{B}}^{s,q,L}_{p,t,r,\omega} (X)$
  14. Atomic decompositions for $\dot{\mathcal{F}}^{s,q,L}_{p,t,r,\omega} (X) .$
  15. Molecular decompositions
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $\mu (X) =\infty$. There exists a collection of open sets $\{ Q_\tau ^k \subset X : k \in \mathbb Z, \tau \in I_k \}$, where $I_k$ denotes certain index set depending on $k$ and constants $\gamma \in (0,1)$, $a_0 \in (0,1]$ and $\kappa_0 \in (0,\infty)$ such that

Theorems & Definitions (81)

  • Lemma 2.1
  • Remark 2.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3: Lemma 3.2, BDL18
  • Definition 2.2
  • Lemma 2.4: Proposition 2.11, BBD20
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 71 more