论文标题
Calderón-Zygmund操作员在同质类型的空间上有两个重量不平等
A two weight inequality for Calderón-Zygmund operators on spaces of homogeneous type with applications
论文作者
论文摘要
让$（x，d，μ）$成为Coifman和Weiss的同质类型的空间，即$ D $是$ x $的准公制，$ x $，$μ$是满足双倍条件的积极措施。假设$ u $和$ v $是$（x，d，μ）$的两个本地有限的正骨措施。在满足侧面条件的一对权重的前提下，我们表征了$ l^{2}（u）$ to $ l^{2}（v）$的calderón-zygmund操作员$ t $ $ t $。对于每个Cube $ b \ subset x $，我们都有以下测试条件，$ \ mathbf {1} _ {b} $作为$ b $ \ begin {equation*} \ vert t（u \ mathbf {1} _ {1} _ {b} _ {b} _ {b} _ {b} _ {b} _ {b} _ {b}） \ Mathcal {t} \ Vert 1_ {B} \ Vert _ {l^{2}（u）}，\ end {equation*} \ begin {equication*} \ vert t^{\ ast} u）} \ leq \ mathcal {t} \ vert 1_ {b} \ vert _ {l^{2}（v）}。 \ end {等式*}证明使用停止立方体和电晕分解，源自纳扎罗夫，Treil和Volberg的作品以及关键的侧面条件。
Let $(X,d,μ)$ be a space of homogeneous type in the sense of Coifman and Weiss, i.e. $d$ is a quasi metric on $X$ and $μ$ is a positive measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite positive Borel measures on $(X,d,μ)$. Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderón--Zygmund operator $T$ from $L^{2}(u)$ to $L^{2}(v)$ in terms of the $A_{2}$ condition and two testing conditions. For every cube $B\subset X$, we have the following testing conditions, with $\mathbf{1}_{B}$ taken as the indicator of $B$ \begin{equation*} \Vert T(u\mathbf{1}_{B})\Vert _{L^{2}(B, v)}\leq \mathcal{T}\Vert 1_{B}\Vert _{L^{2}(u)}, \end{equation*} \begin{equation*} \Vert T^{\ast }(v\mathbf{1}_{B})\Vert _{L^{2}(B, u)}\leq \mathcal{T}\Vert 1_{B}\Vert _{L^{2}(v)}. \end{equation*} The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.
