$L^p$-$L^q$ Boundedness of Spectral Multipliers of the Anharmonic Oscillator

Marianna Chatzakou; Vishvesh Kumar

doi:10.5802/crmath.290

Harmonic analysis

L^{p}

-

L^{q}

Boundedness of Spectral Multipliers of the Anharmonic Oscillator

Marianna Chatzakou ¹; Vishvesh Kumar ¹

¹ Department of Mathematics: Analysis Logic and Discrete Mathematics, Ghent University, Belgium

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 343-347.

Abstract (Original language)
Résumé

In this note we study the $L^{p} - L^{q}$ boundedness of Fourier multipliers of anharmonic oscillators, and as a consequence also of spectral multipliers, for the range $1 < p \leq 2 \leq q < \infty$ . The underlying Fourier analysis is associated with the eigenfunctions of an anharmonic oscillator in some family of differential operators having derivatives of any order. Our analysis relies on a version of the classical Paley-type inequality, introduced by Hörmander, that we extend in our nonharmonic setting.

Dans cette note, nous étudions la $L^{p}$ - $L^{q}$ continuité des multiplicateurs de Fourier des oscillateurs anharmoniques, et par conséquent des multiplicateurs spectraux également, pour $1 < p \leq 2 \leq q < \infty$ . L’analyse de Fourier sous-jacente est associée aux fonctions propres d’un oscillateur anharmonique dans certaines familles d’opérateurs différentiels ayant des dérivées d’ordre quelconque. Notre analyse s’appuie sur une version de l’inégalité classique de type Paley, introduite par Hörmander, que nous étendons dans notre cadre non harmonique.

Received: 2021-05-10
Revised: 2021-10-24
Accepted: 2021-10-24
Published online: 2022-04-26

DOI: 10.5802/crmath.290

Classification: 42B15, 58J40, 47B10, 47G30, 35P10

Author's affiliations:

Marianna Chatzakou ¹; Vishvesh Kumar ¹

¹ Department of Mathematics: Analysis Logic and Discrete Mathematics, Ghent University, Belgium

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2022__360_G4_343_0,
     author = {Marianna Chatzakou and Vishvesh Kumar},
     title = {$L^p$-$L^q$ {Boundedness} of {Spectral} {Multipliers} of the {Anharmonic} {Oscillator}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {343--347},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.290},
     language = {en},
}

TY  - JOUR
AU  - Marianna Chatzakou
AU  - Vishvesh Kumar
TI  - $L^p$-$L^q$ Boundedness of Spectral Multipliers of the Anharmonic Oscillator
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 343
EP  - 347
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.290
LA  - en
ID  - CRMATH_2022__360_G4_343_0
ER  -

%0 Journal Article
%A Marianna Chatzakou
%A Vishvesh Kumar
%T $L^p$-$L^q$ Boundedness of Spectral Multipliers of the Anharmonic Oscillator
%J Comptes Rendus. Mathématique
%D 2022
%P 343-347
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.290
%G en
%F CRMATH_2022__360_G4_343_0

Marianna Chatzakou; Vishvesh Kumar. $L^p$-$L^q$ Boundedness of Spectral Multipliers of the Anharmonic Oscillator. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 343-347. doi : 10.5802/crmath.290. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.290/

References
Cited by

[1] Rauan Kh. Akylzhanov; Erlan Nursultanov; Michael Ruzhansky Hardy–Littlewood–Paley inequalities and Fourier multipliers on $S U (2)$ , Stud. Math., Volume 236 (2016) no. 1, pp. 1-29 | DOI | MR | Zbl

[2] Rauan Kh. Akylzhanov; Erlan Nursultanov; Michael Ruzhansky Hardy–Littlewood, Hausdorff–Young–Paley inequalities, and $L^{p}$ - $L^{q}$ Fourier multipliers on compact homogeneous manifolds, J. Math. Anal. Appl., Volume 479 (2019) no. 2, pp. 1519-1548 | DOI | MR | Zbl

[3] Rauan Kh. Akylzhanov; Michael Ruzhansky $L^{p} - L^{q}$ multipliers on locally compact groups, J. Funct. Anal., Volume 278 (2019) no. 3, 108324 | MR | Zbl

[4] Richard Beals $L^{p}$ Hölder estimates for pseudo-differential operators: sufficient conditions, Ann. Inst. Fourier, Volume 29 (1979) no. 3, pp. 239-260 | DOI | Zbl

[5] Anthony Carbery; Andreas Seeger $H^{p}$ and $L^{p}$ -Variants of multiparameter Calderón–Zygmund theory, Trans. Am. Math. Soc., Volume 334 (1992) no. 2, pp. 719-747 | Zbl

[6] Marianna Chatzakou; Julio Delgado; Michael Ruzhansky On a class of anharmonic oscillators (2020) (https://arxiv.org/abs/1811.12566v3, to appear in Journal de Mathématiques Pures et Appliquées)

[7] Marianna Chatzakou; Vishvesh Kumar $L^{p} - L^{q}$ boundedness of Fourier multipliers associated with the anharmonic Oscillator (2020) (https://arxiv.org/abs/2004.07801v1)

[8] Michael G. Cowling; Saverio Giulini; Stefano Meda $L^{p} - L^{q}$ estimates for functions of theLaplace–Beltrami operator on noncompact symmetric spaces. I, Duke Math. J., Volume 72 (1993) no. 1, pp. 109-150 | Zbl

[9] Bernard Helffer; Didier Robert Comportement asymptotique précise du spectre d’opérateurs globalement elliptiques dans $ℝ^{n}$ , Goulaouic-Meyer-Schwartz Seminar, 1980–1981, École Polytech., Palaiseau, 1981, II | Zbl

[10] Lars Hörmander Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math., Volume 104 (1960), pp. 93-140 | DOI | MR | Zbl

[11] Solomon G. Mikhlin On multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR, Volume 109 (1956), pp. 73-84 | MR | Zbl

[12] Michael Ruzhansky; Niyaz Tokmagambetov Nonharmonic analysis of boundary value problems, Int. Math. Res. Not., Volume 12 (2016), pp. 3548-3615 | DOI | Zbl

[13] Duván C. Sánchez; Vishvesh Kumar; Michael Ruzhansky; Niyaz Tokmagambetov $L^{p} - L^{q}$ boundedness of pseudo-differential operators on smooth manifolds and its applications to nonlinear equations (2020) (https://arxiv.org/abs/2005.04936)

Cited by Sources:

Comments - Policy