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

Chatzakou, Marianna; Kumar, Vishvesh

Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 343-347.

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 $1<p\le 2\le q<\infty$. 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.

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 $1<p\le 2\le q<\infty$. 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.

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

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