Lower bounds for the Dyadic Hilbert transform

Philippe Jaming; Elodie Pozzi; Brett D. Wick

doi:10.5802/afst.1569

Philippe Jaming ¹ ; Elodie Pozzi ¹ ; Brett D. Wick ²

¹ Univ. Bordeaux, IMB, CNRS UMR 5251, 33400 Talence, France
² Department of Mathematics, Washington University – St. Louis, One Brookings Drive, St. Louis, MO 63130-4899, USA

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 1, pp. 265-284.

Résumé
Abstract

Dans cet article, nous établissons des bornes pour la transformée de Hilbert dyadique (Haar shift) de la forme ${∥Ш f∥}_{L^{2} (K)} \geq C (I, K) {∥f∥}_{L^{2} (I)}$ où $I$ et $K$ sont des intervalles dyadiques et $f$ est à support dans $I$ . Si $I \subset K$ de telles bornes existent sans condition supplémentaire sur $f$ alors que dans les cas $K ⊊ I$ et $K \cap I = \emptyset$ une telle borne n’existe que si on impose une condition sur la dérivée de $f$ . Dans le dernier cas nous établissons une borne de la forme ${∥Ш f∥}_{L^{2} (K)} \geq C (I, K) | {〈f〉}_{I} |$ où ${〈f〉}_{I}$ est la moyenne de $f$ sur $I$ . Ce travail permet ainsi une meilleure compréhension du problème similaire pour la transformée de Hilbert sur $ℝ$ .

In this paper, we seek lower bounds of the dyadic Hilbert transform (Haar shift) of the form ${∥Ш f∥}_{L^{2} (K)} \geq C (I, K) {∥f∥}_{L^{2} (I)}$ where $I$ and $K$ are two dyadic intervals and $f$ supported in $I$ . If $I \subset K$ , such bounds exist while in the other cases $K ⊊ I$ and $K \cap I = \emptyset$ such bounds are only available under additional constraints on the derivative of $f$ . In the later case, we establish a bound of the form ${∥Ш f∥}_{L^{2} (K)} \geq C (I, K) | {〈f〉}_{I} |$ where ${〈f〉}_{I}$ is the mean of $f$ over $I$ . This sheds new light on the similar problem for the usual Hilbert transform.

Reçu le : 2016-06-01
Accepté le : 2016-10-15
Publié le : 2018-02-21

MR Zbl

DOI : 10.5802/afst.1569

Classification : 42B20
Mots-clés : Dyadic Hilbert transform, Haar Shift, BMO

Affiliations des auteurs :

Philippe Jaming ¹ ; Elodie Pozzi ¹ ; Brett D. Wick ²

¹ Univ. Bordeaux, IMB, CNRS UMR 5251, 33400 Talence, France
² Department of Mathematics, Washington University – St. Louis, One Brookings Drive, St. Louis, MO 63130-4899, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Philippe Jaming and Elodie Pozzi and Brett D. Wick},
     title = {Lower bounds for the {Dyadic} {Hilbert} transform},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {265--284},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
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     year = {2018},
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     language = {en},
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Philippe Jaming; Elodie Pozzi; Brett D. Wick. Lower bounds for the Dyadic Hilbert transform. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 1, pp. 265-284. doi : 10.5802/afst.1569. https://afst.centre-mersenne.org/articles/10.5802/afst.1569/

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