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We prove Hilbert transform identities involving conformal maps via the use of Rellich identity and the solution of the Neumann problem in a graph Lipschitz domain in the plane. We obtain as consequences new $L^2$-weighted estimates for the Hilbert transform, including a sharp bound for its norm as a bounded operator in weighted $L^2$ in terms of a weight constant associated to the Helson-Szeg\"o theorem.
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On the Maximal Directional Hilbert Transform in Three Dimensions
Abstract We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $$H_\Omega $$ along finite subsets of a finite order lacunary set of directions $$\Omega \subset \mathbb{R}^3$$, answering a question of Parcet and Rogers in dimension $n=3$. Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2. The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of 2D angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting.
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- Award ID(s):
- 2000510
- PAR ID:
- 10181135
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2020
- Issue:
- 14
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 4324 to 4356
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rny138
