Commutators of Hilbert transforms along monomial curves
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2019
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| _version_ | 1866917771936268288 |
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| author | Bongers, Rosemarie Guo, Zihua Li, Ji Wick, Brett D. |
| author_facet | Bongers, Rosemarie Guo, Zihua Li, Ji Wick, Brett D. |
| contents | The Hilbert transforms associated with monomial curves have a natural non-isotropic structure. We study the commutator of such Hilbert transforms and a symbol $b$ and prove the upper bound of this commutator when $b$ is in the corresponding non-isotropic BMO space by using the Cauchy integral trick. We also consider the lower bound of this commutator by introducing a new testing BMO space associated with the given monomial curve, which shows that the classical non-isotropic BMO space is contained in the testing BMO space. We also show that the non-zero curvature of such monomial curves are important, since when considering Hilbert transforms associated with lines, the parallel version of non-isotropic BMO space and testing BMO space have overlaps but do not have containment. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_02118 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Commutators of Hilbert transforms along monomial curves Bongers, Rosemarie Guo, Zihua Li, Ji Wick, Brett D. Classical Analysis and ODEs The Hilbert transforms associated with monomial curves have a natural non-isotropic structure. We study the commutator of such Hilbert transforms and a symbol $b$ and prove the upper bound of this commutator when $b$ is in the corresponding non-isotropic BMO space by using the Cauchy integral trick. We also consider the lower bound of this commutator by introducing a new testing BMO space associated with the given monomial curve, which shows that the classical non-isotropic BMO space is contained in the testing BMO space. We also show that the non-zero curvature of such monomial curves are important, since when considering Hilbert transforms associated with lines, the parallel version of non-isotropic BMO space and testing BMO space have overlaps but do not have containment. |
| title | Commutators of Hilbert transforms along monomial curves |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/1909.02118 |