Harmonic analysis
L p -L q Boundedness of Spectral Multipliers of the Anharmonic Oscillator
Comptes Rendus. Mathématique, Volume 360 (2022) no. G4, pp. 343-347.

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<p2q<. 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<p2q<. 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:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.290
Classification: 42B15, 58J40, 47B10, 47G30, 35P10
Chatzakou, Marianna 1; Kumar, Vishvesh 1

1 Department of Mathematics: Analysis Logic and Discrete Mathematics, Ghent University, Belgium
@article{CRMATH_2022__360_G4_343_0,
     author = {Chatzakou, Marianna and Kumar, Vishvesh},
     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},
     number = {G4},
     year = {2022},
     doi = {10.5802/crmath.290},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.290/}
}
TY  - JOUR
AU  - Chatzakou, Marianna
AU  - Kumar, Vishvesh
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
IS  - G4
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.290/
DO  - 10.5802/crmath.290
LA  - en
ID  - CRMATH_2022__360_G4_343_0
ER  - 
%0 Journal Article
%A Chatzakou, Marianna
%A Kumar, Vishvesh
%T $L^p$-$L^q$ Boundedness of Spectral Multipliers of the Anharmonic Oscillator
%J Comptes Rendus. Mathématique
%D 2022
%P 343-347
%V 360
%N G4
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.290/
%R 10.5802/crmath.290
%G en
%F CRMATH_2022__360_G4_343_0
Chatzakou, Marianna; Kumar, Vishvesh. $L^p$-$L^q$ Boundedness of Spectral Multipliers of the Anharmonic Oscillator. Comptes Rendus. Mathématique, Volume 360 (2022) no. G4, pp. 343-347. doi : 10.5802/crmath.290. http://www.numdam.org/articles/10.5802/crmath.290/

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

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

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

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

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

[8] Cowling, Michael G.; Giulini, Saverio; Meda, Stefano 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] Helffer, Bernard; Robert, Didier Comportement asymptotique précise du spectre d’opérateurs globalement elliptiques dans n , Goulaouic-Meyer-Schwartz Seminar, 1980–1981, École Polytech., Palaiseau, 1981, II | Zbl

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

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

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

[13] Sánchez, Duván C.; Kumar, Vishvesh; Ruzhansky, Michael; Tokmagambetov, Niyaz 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: