Littlewood–Paley–Stein functionals: an ${\mathcal{R}}$-boundedness approach

Cometx, Thomas; Ouhabaz, El Maati

doi:10.5802/aif.3634

Littlewood–Paley–Stein functionals: an

ℛ

-boundedness approach
[Fonctionnelles de Littlewood–Paley–Stein : une approche par la

ℛ

-bornétude]

Cometx, Thomas ¹ ; Ouhabaz, El Maati ²

¹ IMB, CNRS and Univ. Bordeaux, 351 Cours de la Libération, 33405 Talence (France)
² IMB, CNRS, Univ. Bordeaux, 351 Cours de la Libération, 33405 Talence (France)

Annales de l'Institut Fourier, Online first, 46 p.

Résumé
Abstract

Soit $L = Δ + V$ un opérateur de Schrödinger avec un potentiel positif $V$ sur une variété Riemannianne complète $M$ . Nous montrons que les fonctionnelles verticales de Littlewood–Paley–Stein associées à $L$ sont bornées sur $L^{p} (M)$ si et seulement si l’ensemble ${\sqrt{t} \nabla e^{- t L}, t > 0}$ est $ℛ$ -borné sur $L^{p} (M)$ . Nous introduisons et étudions d’autres fonctionnelles plus générales. Pour une suite de fonctions $m_{k} : [0, \infty) \to ℂ$ données, on définit

H ((f_{k})) : = {(\sum_{k} \int_{0}^{\infty} {| \nabla m_{k} (t L) f_{k} |}^{2} d t)}^{1 / 2} + {(\sum_{k} \int_{0}^{\infty} {| \sqrt{V} m_{k} (t L) f_{k} |}^{2} d t)}^{1 / 2} .

Nous montrons que $H$ est bornée sur $L^{p} (M)$ au sens

∥ H ((f_{k})) ∥_{p} \leq C {∥{(\sum_{k} {| f_{k} |}^{2})}^{1 / 2}∥}_{p}

avec une constante $C$ indépendante de ${(f_{k})}_{k}$ . Une estimation inférieure est aussi démontrée sur l’espace dual $L^{p^{'}}$ . Nous discuterons le lien entre ces fonctionnelles et la transformée de Riesz. Plusieurs exemples et contre-exemples sont donnés dans le papier.

Let $L = Δ + V$ be a Schrödinger operator with a non-negative potential $V$ on a complete Riemannian manifold $M$ . We prove that the vertical Littlewood-Paley-Stein functional associated with $L$ is bounded on $L^{p} (M)$ if and only if the set ${\sqrt{t} \nabla e^{- t L}, t > 0}$ is $ℛ$ -bounded on $L^{p} (M)$ . We also introduce and study more general functionals. For a sequence of functions $m_{k} : [0, \infty) \to ℂ$ , we define

H ((f_{k})) : = {(\sum_{k} \int_{0}^{\infty} {| \nabla m_{k} (t L) f_{k} |}^{2} d t)}^{1 / 2} + {(\sum_{k} \int_{0}^{\infty} {| \sqrt{V} m_{k} (t L) f_{k} |}^{2} d t)}^{1 / 2} .

We prove boundedness of $H$ on $L^{p} (M)$ in the sense

∥ H ((f_{k})) ∥_{p} \leq C {∥{(\sum_{k} {| f_{k} |}^{2})}^{1 / 2}∥}_{p}

for some constant $C$ independent of ${(f_{k})}_{k}$ . A lower estimate is also proved on the dual space $L^{p^{'}}$ . We introduce and study boundedness of other Littlewood-Paley-Stein type functionals and discuss their relationships to the Riesz transform. Several examples are given in the paper.

Reçu le : 2020-12-09
Révisé le : 2022-09-19
Accepté le : 2022-11-28
Première publication : 2024-05-17

DOI : 10.5802/aif.3634

Classification : 42B25, 58J35, 47D08, 60G50
Keywords: Littlewood–Paley–Stein functionals, Riesz transforms, Kahane–Khintchine inequality, spectral multipliers, Schrödinger operators, elliptic operators.
Mot clés : Fonctionnelles de Littlewood–Paley–Stein, transformée de Riesz, inégalités de Kahane–Khintchine, multiplicateurs spectraux, opérateurs de Schrödinger, opérateurs elliptiques.

Affiliations des auteurs :

Cometx, Thomas ¹ ; Ouhabaz, El Maati ²

¹ IMB, CNRS and Univ. Bordeaux, 351 Cours de la Libération, 33405 Talence (France)
² IMB, CNRS, Univ. Bordeaux, 351 Cours de la Libération, 33405 Talence (France)

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     title = {Littlewood{\textendash}Paley{\textendash}Stein functionals: an ${\mathcal{R}}$-boundedness approach},
     journal = {Annales de l'Institut Fourier},
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     language = {en},
     note = {Online first},
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Cometx, Thomas; Ouhabaz, El Maati. Littlewood–Paley–Stein functionals: an ${\mathcal{R}}$-boundedness approach. Annales de l'Institut Fourier, Online first, 46 p.

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