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.

Dans cet article, nous établissons des bornes pour la transformée de Hilbert dyadique (Haar shift) de la forme Шf L 2 (K) C(I,K)f L 2 (I) I et K sont des intervalles dyadiques et f est à support dans I. Si IK de telles bornes existent sans condition supplémentaire sur f alors que dans les cas KI et KI= 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 Шf L 2 (K) C(I,K)|f I |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 .

In this paper, we seek lower bounds of the dyadic Hilbert transform (Haar shift) of the form Шf L 2 (K) C(I,K)f L 2 (I) where I and K are two dyadic intervals and f supported in I. If IK, such bounds exist while in the other cases KI and KI= such bounds are only available under additional constraints on the derivative of f. In the later case, we establish a bound of the form Шf L 2 (K) 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 :
Accepté le :
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DOI : https://doi.org/10.5802/afst.1569
Classification : 42B20
Mots clés : Dyadic Hilbert transform, Haar Shift, BMO
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     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},
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Jaming, Philippe; Pozzi, Elodie; Wick, Brett D. 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. http://archive.numdam.org/articles/10.5802/afst.1569/

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