Asymptotic behavior of $\mathrm {L}^p$ estimates for a class of multipliers with homogeneous unimodular symbols
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- by Aleksandar Bulj and Vjekoslav Kovač;
- Trans. Amer. Math. Soc. 376 (2023), 4539-4567
- DOI: https://doi.org/10.1090/tran/8883
- Published electronically: April 3, 2023
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Abstract:
We study Fourier multiplier operators associated with symbols $\xi \mapsto \exp (\mathbb {i}\lambda \phi (\xi /|\xi |))$, where $\lambda$ is a real number and $\phi$ is a real-valued $\mathrm {C}^\infty$ function on the standard unit sphere $\mathbb {S}^{n-1}\subset \mathbb {R}^n$. For $1<p<\infty$ we investigate asymptotic behavior of norms of these operators on $\mathrm {L}^p(\mathbb {R}^n)$ as $|\lambda |\to \infty$. We show that these norms are always $O((p^\ast -1) |\lambda |^{n|1/p-1/2|})$, where $p^\ast$ is the larger number between $p$ and its conjugate exponent. More substantially, we show that this bound is sharp in all even-dimensional Euclidean spaces $\mathbb {R}^n$. In particular, this gives a negative answer to a question posed by Maz’ya. Concrete operators that fall into the studied class are the multipliers forming the two-dimensional Riesz group, given by the symbols $r\exp (\mathbb {i}\varphi ) \mapsto \exp (\mathbb {i}\lambda \cos \varphi )$. We show that their $\mathrm {L}^p$ norms are comparable to $(p^\ast -1) |\lambda |^{2|1/p-1/2|}$ for large $|\lambda |$, solving affirmatively a problem suggested in the work of Dragičević, Petermichl, and Volberg.References
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Bibliographic Information
- Aleksandar Bulj
- Affiliation: Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
- ORCID: 0000-0003-2751-0636
- Email: aleksandar.bulj@math.hr
- Vjekoslav Kovač
- Affiliation: Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
- MR Author ID: 962691
- Email: vjekovac@math.hr
- Received by editor(s): April 18, 2022
- Published electronically: April 3, 2023
- Additional Notes: This work was supported in part by the Croatian Science Foundation project UIP-2017-05-4129 (MUNHANAP)
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 4539-4567
- MSC (2020): Primary 42B15; Secondary 42B20
- DOI: https://doi.org/10.1090/tran/8883
- MathSciNet review: 4608424