AI 中文总结
该研究围绕可变洛伦兹空间$\mathfrak{L}^{p(\cdot), q(\cdot)}(\mathbb{R}^n)$展开，建立其性质以定义可变哈代 - 洛伦兹空间，推导原子分解，获得奇异积分等算子有界性及分数极大算子的费弗曼 - 斯坦向量值不等式。
AI 中文摘要
本注记旨在进一步建立由L. 埃弗雷米德泽、V. 科基拉什维利和S. 萨姆科引入的可变洛伦兹空间$\mathfrak{L}^{p(\cdot), q(\cdot)}(\mathbb{R}^n)$的性质，以应用与球拟巴拿赫函数空间相关的哈代空间理论，定义与之相关的可变哈代 - 洛伦兹空间。接着可立即推导这些空间的有限和无限原子分解。还得到了可变哈代 - 洛伦兹空间中奇异积分和分数型算子的有界性，并给出了可变洛伦兹空间$r$ - 凸化上分数极大算子的费弗曼 - 斯坦向量值不等式。
英文摘要
The purpose of this note is to establish further properties of the variable Lorentz spaces $\mathfrak{L}^{p(\cdot), q(\cdot)}(\mathbb{R}^n)$ introduced by L. Ephremidze, V. Kokilashvili and S. Samko, which will allow us to apply the theory of Hardy spaces associated with ball quasi-Banach function spaces, and so define the variable Hardy-Lorentz spaces associated with $\mathfrak{L}^{p(\cdot), q(\cdot)}(\mathbb{R}^n)$. Then, the finite and infinite atomic decompositions for these spaces will be deduced immediately. We also obtain the boundedness of singular integrals and fractional type operators in variable Hardy-Lorentz spaces, and provide a Fefferman-Stein vector-valued inequality for the fractional maximal operator on the $r$-convexification of variable Lorentz spaces.
Comments18 pages
