The Clifford Hilbert transform and Riemann–Hilbert problems associated to the Dirac operator in Lebesgue spaces
The Clifford Hilbert transform and Riemann–Hilbert problems associated to the Dirac operator in Lebesgue spaces
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与勒贝格空间中的狄拉克算子相关的克利福德希尔伯特变换和黎曼希尔伯特问题
DOI：
10.1016/j.amc.2019.124686
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发表时间：
2020
影响因子：
4
通讯作者：
Longfei Gu
中科院分区：
文献类型：
--
作者：
Longfei Gu
建立了Clifford分析中Hilbert变换的一些性质，并利用Clifford代数和一维证明的分析推广，给出了欧氏空间Rm（m≥ 3）中Clifford Hilbert变换的弱（1，1）型不等式.此外，我们还证明了Clifford Hilbert变换的Kolmogorov不等式。利用Hilbert变换的性质和Clifford分析技巧，建立了Clifford值Lebesgue p-可积空间中的Riemann-Hilbert问题.基于Newton嵌入方法，证明了非线性Riemann-Hilbert问题解的存在唯一性，并给出了Lebesgue空间中Newton嵌入方法近似解的误差估计.
In this paper, we establish some properties of the Hilbert transform in Clifford analysis setting, and mainly show the weak type (1, 1) inequality for the Clifford Hilbert transform in Euclidean space R m (m≥ 3) using the Clifford algebra and analysis generalizations of the one dimensional proofs. Furthermore, we prove Kolmogorov’s inequality for Clifford Hilbert transform. With the help of the properties for the Hilbert transform and Clifford analytic techniques, we establish the Riemann–Hilbert problems in Clifford value Lebesgue p-integrable spaces. Based on the Newton embedding method, we prove an existence and uniqueness for the nonlinear Riemann–Hilbert problem and also give an error estimation for the approximate solutions in the Newton embedding procedure in Lebesgue spaces.