Bourgain–Morrey Spaces Mixed with Structure of Besov Spaces

Pub Date : 2024-03-06 DOI:10.1134/s0081543823050152

Yirui Zhao, Yoshihiro Sawano, Jin Tao, Dachun Yang, Wen Yuan

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Abstract

Bourgain–Morrey spaces \(\mathcal{M}^p_{q,r}(\mathbb R^n)\), generalizing what was introduced by J. Bourgain, play an important role in the study related to the Strichartz estimate and the nonlinear Schrödinger equation. In this article, via adding an extra exponent \(\tau\), the authors creatively introduce a new class of function spaces, called Besov–Bourgain–Morrey spaces \(\mathcal{M}\dot{B}^{p,\tau}_{q,r}(\mathbb R^n)\), which is a bridge connecting Bourgain–Morrey spaces \(\mathcal{M}^p_{q,r}(\mathbb R^n)\) with amalgam-type spaces \((L^q,\ell^r)^p(\mathbb R^n)\). By making full use of the Fatou property of block spaces in the weak local topology of \(L^{q'}(\mathbb R^n)\), the authors give both predual and dual spaces of \(\mathcal{M}\dot{B}^{p,\tau}_{q,r}(\mathbb R^n)\). Applying these properties and the Calderón product, the authors also establish the complex interpolation of \(\mathcal{M}\dot{B}^{p,\tau}_{q,r}(\mathbb R^n)\). Via fully using fine geometrical properties of dyadic cubes, the authors then give an equivalent norm of \(\|\kern1pt{\cdot}\kern1pt\|_{\mathcal{M}\dot{B}^{p,\tau}_{q,r}(\mathbb R^n)}\) having an integral expression, which further induces a boundedness criterion of operators on \(\mathcal{M}\dot{B}^{p,\tau}_{q,r}(\mathbb R^n)\). Applying this criterion, the authors obtain the boundedness on \(\mathcal{M}\dot{B}^{p,\tau}_{q,r}(\mathbb R^n)\) of classical operators including the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator.

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与贝索夫空间结构相混合的布尔干涉-莫雷空间

摘要布尔干姆-莫雷空间（Bourgain-Morrey spaces）是对布尔干（J. Bourgain）引入的空间的概括，在与斯特里查兹估计和非线性薛定谔方程相关的研究中发挥着重要作用。在本文中，通过添加一个额外的指数（\tau\），作者创造性地引入了一类新的函数空间，即 Besov-Bourgain-Morrey 空间（\mathcal{M}\dot{B}^{p、\(\mathcal{M}^p_{q,r}(\mathbb R^n)\)与汞齐型空间 \((L^q,ell^r)^p(\mathbb R^n)\)之间的桥梁。通过充分利用 \(L^{q'}(\mathbb R^n)\的弱局部拓扑中块空间的法图性质，作者给出了 \(\mathcal{M}\dot{B}^{p,\tau}_{q,r}(\mathbb R^n)\的前空间和对偶空间。）应用这些性质和卡尔德龙积，作者还建立了 \(\mathcal{M}\dot{B}^{p,\tau}_{q,r}(\mathbb R^n)\) 的复插值。通过充分利用二元立方体的精细几何性质，作者给出了一个等价的规范\(|\kern1pt{\cdot}\kern1pt\|_\mathcal{M}\dot{B}^{p,\tau}_{q、r}(\mathbb R^n)}）有一个积分表达式，这进一步引出了一个关于 \(\mathcal{M}\dot{B}^{p,\tau}_{q,r}(\mathbb R^n)}）的算子有界性准则。应用这一准则，作者得到了包括哈代-利特尔伍德最大算子、分数积分和卡尔德龙-齐格蒙算子在内的经典算子在 \(\mathcal{M}\dot{B}^{p,\tau}_{q,r}(\mathbb R^n)\) 上的有界性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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