论文标题
与球类型的空间上与球准 - 浴的功能空间相关的弱耐硬地点
Weak Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Decompositions, Real Interpolation, and Calderón--Zygmund Operators
论文作者
论文摘要
令$（\ mathbb {x}，d，μ）$是R. R. R. Coifman和G. Weiss的均匀类型的一个空间，以及$ x（\ Mathbb {x}）$ a ball quasi-banach函数在$ \ mathbb {x} $上的球。在本文中，作者介绍了与$ x（\ Mathbb {x}）$关联的弱hardy空间$ WH_X（\ Mathbb {x}）$通过Grand Maximal功能相关联，并通过其他最大功能和ATOMS来表征$ WH_X（\ Mathbb {x}）$。然后，作者在关键案例中应用这些特征来获得Calderón-Zygmund操作员的实际插值和界限。本文的主要新颖性存在于作者使用aoki-rolewicz定理以及二元系统和身份近似值的指数衰减，这些近似值与$ \ \ \ m i \ mathbb {x} $的几何属性紧密相连，以克服困难，从而克服了这种困难。 $ \ | \ cdot \ | _ {x（\ Mathbb {x}）} $以及正在考虑的量度$μ$的反向加倍假设，还使用$ x（\ m athbb {x}）$ x（\ mathbb {x}）$与弱的空间$ wx（$ wx（$ wx bbb）$ wx（x Mathbb {x）$ to $ x（x} $ x）$ x（x} $ x）证明原子的无限求和在$ \ mathbb {x} $上的分布空间中收敛。此外，所有这些结果具有广泛的一般性，尤其是，即使它们应用于加权的Lebesgue空间，Orlicz空间和可变的Lebesgue空间时，获得的结果也是新的，实际上，其中有些甚至是RD空间，即使在RD空间上也是新的（即，同质类型的空间，都可以满足额外的反向加倍的调整）。
Let $(\mathbb{X},d,μ)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space $WH_X(\mathbb{X})$ associated with $X(\mathbb{X})$ via the grand maximal function, and characterize $WH_X(\mathbb{X})$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón--Zygmund operators in the critical case. The main novelties of this article exist in that the authors use the Aoki--Rolewicz theorem and both the dyadic system and the exponential decay of approximations of the identity on $\mathbb{X}$, which closely connect with the geometrical properties of $\mathbb{X}$, to overcome the difficulties caused by the absence of both the triangle inequality of $\|\cdot\|_{X(\mathbb{X})}$ and the reverse doubling assumption of the measure $μ$ under consideration, and also use the relation between the convexification of $X(\mathbb{X})$ and the weak space $WX(\mathbb{X})$ associated with $X(\mathbb{X})$ to prove that the infinite summation of atoms converges in the space of distributions on $\mathbb{X}$. Moreover, all these results have a wide range of generality and, particularly, even when they are applied to the weighted Lebesgue space, the Orlicz space, and the variable Lebesgue space, the obtained results are also new and, actually, some of them are new even on RD-spaces (namely, spaces of homogeneous type satisfying the additional reverse doubling condition).
