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An estimate on the Bedrosian commutator in Sobolev space
- Marcel Oliver^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-018-1940-3
© The Author(s) 2018
- Received: 28 August 2018
- Accepted: 7 December 2018
- Published: 17 December 2018
Abstract
While the Bedrosian identity for the Hilbert transform of a product does not hold for general Sobolev class functions, we show that the defect of this identity is more regular than would be naively expected. We use this result to give a stronger-than-expected estimate on the chain rule defect of the square root of the Laplacian.
Keywords
- Bedrosian identity
- Sobolev space
- Banach algebra
MSC
- 26D10
- 42A85
- 46E35
1 Introduction
The Bedrosian identity usually arises in the context of time-frequency analysis. A key question there is the notion of instantaneous amplitude and frequency, where the instantaneous frequency is the derivative of the phase function and is required to be non-negative. Signals with this property are called mono-components; arbitrary signals may be decomposed into mono-components in various ways. When the phase signal is already a mono-component, then its product with an amplitude function is also a mono-component if and only if the Bedrosian identity with u as amplitude and v as phase function holds [5, 6]. This observation led to a quest to characterize necessary and sufficient conditions for the Bedrosian identity [3, 9, 10]. A full characterization of mono-components, which includes mono-components of non-Bedrosian type, is given in [7]. For computational approaches on signal decompositions into mono-components, see, e.g., [4, 8].
In this note, we are asking a very different question. Given that the conditions under which the Bedrosian identity holds are highly non-generic, is it possible to provide a generic estimate on the defect that is stronger than separate estimates on the left- and right-hand sides of (1)? In the following, we shall show that this is indeed the case when estimating the defect—the Bedrosian commutator—in Sobolev space. In the next section, we state and prove this surprisingly simple result. In the final Sect. 3, we show that this result implies that the square root of the Laplacian on \(\mathbb {R}\) satisfies the usual chain rule with a defect that is smoother than each of the separate terms.
2 The commutator estimate
Theorem 1
Proof
Remark
3 Application
Declarations
Acknowledgements
Not applicable.
Availability of data and materials
Not applicable.
Funding
The result arose from work done as part of German Research Foundation funded project OL-155/6-1. The author also thanks the German Research Foundation Collaborative Research Center TRR 181 for funding during the preparation of the manuscript.
Authors’ contributions
The paper is entirely the original work of the author. All authors read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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