论文标题
通过谐波扩展，分数差分运算符的换向器的估计值
Estimates for commutators of fractional differential operators via harmonic extension
论文作者
论文摘要
该硕士论文基于Lenzmann和Schikorra的论文“通过谐波扩展估算的尖锐换向器估计”，在该论文中，他们提出了一种方法来证明涉及Riesz Transforms，分数Laplacians和Riesz电位的换向器的方法，请参见ARXIV：ARXIV：1609.08547。这些证据仅涉及到上半空间的谐波扩展以及在某些基本转移范围旁边的部分集成，因为较深的理论集中在各种可以用作黑盒的痕量表征结果中。 在本文的上半年，在Caffarelli和Silvestre收集了一些基本结果为S骨扩展的一些基本结果之后，我们使用这种方法来证明各种换向器估计值，除了缩短了一些证明，但在Lenzmann和Schikorra之后紧随其后。 在下半年，我们证明了Lenzmann和Schikorra列出的BlackBox估计值的广义版本，并讨论了构成这些BlackBox估计值的不同构建块，包括Triebel-Lizorkin和Besov-Lipschitz空间表征以及方形函数估计值。
This master thesis is based on the paper "Sharp commutator estimates via harmonic extensions" by Lenzmann and Schikorra, in which they proposed a method to prove estimates for commutators involving Riesz transforms, fractional Laplacians and Riesz potentials, see arXiv:1609.08547. These proofs only involve harmonic extensions to the upper half-space and integration by parts next to some elementary transfromations, since the deeper theory is concentrated in a variety of trace characterization results which can be used as a blackbox. In the first half of this thesis, after collecting some elementary results for the s-harmonic extension by Caffarelli and Silvestre, we use this method to prove a variety of commutator estimates, closely following Lenzmann and Schikorra except for shortening some proofs. In the second half, we prove generalized versions of the blackbox estimates listed by Lenzmann and Schikorra and discuss the different building blocks which make up these blackbox estimates, including Triebel-Lizorkin and Besov-Lipschitz space characterizations as well as square function estimates.
