Multi‐parameter estimates via operator‐valued shifts
Multi‐parameter estimates via operator‐valued shifts
复制标题
通过操作员值偏移进行多参数估计
DOI：
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发表时间：
2017
影响因子：
1.8
通讯作者：
Emil Vuorinen
中科院分区：
文献类型：
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作者：
T. Hytönen;Henri Martikainen;Emil Vuorinen
证明了作用于向量值(UMD Banach格值)函数的一类具有多参数奇性的一般奇异积分算子的混合范数LpLq有界性.此外，具有一致假设的这类算子族不仅是一致有界的，而且是R-有界的，这是应用中经常需要的一个真正更强的性质。以前这种性质的结果只涉及卷积型或稍微更一般的无半积奇异积分。相反，我们的分析专门针对多参数设置中出现的不同部分副积的阵列，将它们解释为仿积值单参数运算符。这种新的观点在已有的多参数算子表示结果的基础上提供了概念上的简化，这是证明这些算子有界性的关键。
We prove the mixed‐norm LpLq ‐boundedness of a general class of singular integral operators having a multi‐parameter singularity and acting on vector‐valued (UMD Banach lattice‐valued) functions. Moreover, families of such operators with uniform assumptions are shown to be not only uniformly bounded but R ‐bounded, a genuinely stronger property that is often needed in applications. Previous results of this nature only dealt with convolution‐type or slightly more general paraproduct‐free singular integrals. In contrast, our analysis specifically targets the array of different partial paraproducts that arise in the multi‐parameter setting by interpreting them as paraproduct‐valued one‐parameter operators. This new point‐of‐view provides a conceptual simplification over the existing representation results for multi‐parameter operators, which is a key to the proof of the boundedness of these operators.