论文标题
Köthe-Herz空间：无限直接总和的汞合金型空间
Köthe-Herz Spaces: The Amalgam-Type Spaces of Infinite Direct Sums
论文作者
论文摘要
在本文中，我们介绍了一个称为köthe-herz空间$ e（\ mathcal {x}）$的功能空间。这些空间与amalgam空间相似，其特征是由可数家族$ \ Mathcal {x} = \ left（x_ {α} \ right）_ {α\ in i} $ quasi-normed函数空间的i} $ e $ E $ e $ e $ e $ e $ e $ e $ e $ e $ e quasi-norm-norm-norm-norm-norm-norm sequence sequence sequence sequence sequence semence sequence semence sequence sequence semence semence sequence。我们从其组件中$ e（\ Mathcal {x}）$继承的各种几何和拓扑属性，例如在抽象设置中其完整性，二元性，二元性，订单连续性，理想和FATOU属性。此外，我们为$ e（\ Mathcal {x}）$提供Banach功能空间表征，它使我们能够更深入地了解其结构和行为。此外，通过将Lorentz空间（Orlicz空间）和Lebesgue序列空间的适当合并，我们将Lorentz-Herz空间（Orlicz-Herz空间）定义为$ E（\ Mathcal {X}）$的特定情况，这些情况仍然是经典赫尔兹空间的概括。在这种情况下（尤其是Lorentz-Herz空间），我们建立了先前研究的属性，证明了插值结果，并证明了重要的sublinear积分算子与满足大小条件的内核的界限。
In this paper, we introduce a class of function spaces called Köthe-Herz spaces $E(\mathcal{X})$. These spaces are similar to amalgam spaces and are characterized by a local component given by a countable family $\mathcal{X}=\left( X_{α}\right) _{α\in I}$ of quasi-normed function spaces, and a global component $E$, which is a quasi-normed sequence space. We investigate various geometric and topological properties inherited by $E(\mathcal{X})$ from its components, such as their completeness, duality, order continuity, ideal and Fatou properties, in an abstract setting. In addition, we provide a Banach function space characterization for $E(\mathcal{X})$, which allows us to understand its structure and behavior more deeply. Furthermore, by appropriate amalgamation of Lorentz spaces (Orlicz spaces) and Lebesgue sequence spaces, we define Lorentz-Herz spaces (Orlicz-Herz spaces) as a particular case of $E(\mathcal{X})$, which are still generalizations of the classical Herz spaces. In this context (especially Lorentz-Herz spaces), we establish previously studied properties, demonstrate interpolation results, and prove the boundedness of important sublinear integral operators with kernels that satisfy a size condition.
