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Mixed Bourgain-Morrey spaces and their applications to boundedness of operators

Tengfei Bai, Pengfei Guo, Jingshi Xu

Abstract

We introduce the mixed Bourgain-Morrey spaces and obtain their preduals. The boundedness of Hardy-Littlewood maximal operator, iterated maximal operator, fractional integral operator, singular integral operator on these spaces is proved. In addition, we give a description of the dual of mixed Bourgain-Morrey spaces and conclude the reflexivity of these spaces.

Mixed Bourgain-Morrey spaces and their applications to boundedness of operators

Abstract

We introduce the mixed Bourgain-Morrey spaces and obtain their preduals. The boundedness of Hardy-Littlewood maximal operator, iterated maximal operator, fractional integral operator, singular integral operator on these spaces is proved. In addition, we give a description of the dual of mixed Bourgain-Morrey spaces and conclude the reflexivity of these spaces.
Paper Structure (13 sections, 55 theorems, 188 equations)

This paper contains 13 sections, 55 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of mixed Bourgain-Morrey spaces
  4. Fundamental properties
  5. Nontriviality
  6. Approximation, density and separability
  7. Preduals of mixed Bourgain-Morrey spaces
  8. Properties of block spaces
  9. Operators on mixed Bourgain-Morrey spaces and their preduals
  10. The Hardy-Littlewood maximal operator on mixed Bourgain-Morrey spaces
  11. The Hardy-Littlewood maximal operator on block spaces
  12. Fractional integral operator
  13. Singular integral operators

Key Result

Lemma 2.5

Let $0 < \vec{p} \le \infty$. Let $\{ f_k\}_{k=1}^\infty$ be a sequence of non-negative measurable functions on ${{{\mathbb R}}^n}$. Then $\left\| \liminf _{k\to \infty} f_k \right\|_{L^{\vec{p}}} \le \liminf _{k\to \infty} \left\| f_k \right\|_{L^{\vec{p}}} .$

Theorems & Definitions (122)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • Lemma 2.7
  • Lemma 2.8: Theorem 1, BP61
  • Definition 2.9
  • Lemma 3.1
  • ...and 112 more