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A restriction estimate for a class of oscillatory integral operators along paraboloid
- Shaozhen Xu^{1, 3} and
- Wenjuan Li^{2}Email author
https://doi.org/10.1186/s13660-019-1963-4
© The Author(s) 2019
- Received: 12 June 2018
- Accepted: 10 January 2019
- Published: 23 January 2019
Abstract
Keywords
- Restriction estimate
- Oscillatory integral operator
- Multidimensional Van der Corput
- Knapp’s counterexample
MSC
- 42B20
- 47G10
1 Introduction
The Fourier restriction conjecture has attracted a lot of attention in the development of modern harmonic analysis. It was raised by Stein in the 1960s and can be stated as follows (in its dual form).
Conjecture 1
Theorem 1
2 Proof of Theorem 1
Proof
Lemma 1
Lemma 2
- 1.
\(R_{a}^{b}\) is of weak type \((1,b/a)\).
- 2.
\(R_{a}^{b}\) is bounded from \(L^{p}(\mathbb{R}^{d})\) to \(L^{q}( \mathbb{R}^{d})\), whenever \(\frac{1}{p}=\frac{1}{q}+\frac{b-a}{b}\) and \(1< p<\frac{b}{b-a}\).
3 The necessary condition to guarantee (7)
In this part, we will construct an example in the spirit of Knapp’s counterexample to show the necessary condition of (7). Then we have following.
Theorem 2
Proof
Remark 2
Declarations
Funding
The second author is supported by National Natural Science Foundation of China (No. 11601427), The Fundamental Research Funds for the Central Universities (No. 3102017zy035).
Authors’ contributions
All authors worked in coordination and contributed equally. All authors carried out the proof, read and approved the final version of the manuscript.
Competing interests
The authors declare that they have 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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