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, Serie 6, Volume 27 (2018) no. 1, pp. 265-284.

Abstract
Abstract

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.

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 $ℝ$ .

Received: 2016-06-01
Accepted: 2016-10-15
Published online: 2018-02-21

MR Zbl

DOI: 10.5802/afst.1569

Classification: 42B20
Keywords: Dyadic Hilbert transform, Haar Shift, BMO

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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},
     number = {1},
     year = {2018},
     doi = {10.5802/afst.1569},
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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, Serie 6, Volume 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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