Sharp inequalities for one-sided Muckenhoupt weights

  1. Paul Alton Hagelstein
  2. Ioannis Parissis
  3. Olli Saari
Revue:
Collectanea mathematica

ISSN: 0010-0757

Année de publication: 2018

Volumen: 69

Fascículo: 1

Pages: 151-161

Type: Article

DOI: 10.1007/S13348-017-0201-Y DIALNET GOOGLE SCHOLAR lock_openAccès ouvert editor

D'autres publications dans: Collectanea mathematica

Résumé

Let �+∞ denote the class of one-sided Muckenhoupt weights, namely all the weights w for which �+:��(�)→��,∞(�) for some �>1 , where �+ is the forward Hardy–Littlewood maximal operator. We show that �∈�+∞ if and only if there exist numerical constants �∈(0,1) and �>0 such that �({�∈ℝ:�+1�(�)>�})≤��(�) for all measurable sets �⊂ℝ . Furthermore, letting �+�(�):=sup0<�(�)<+∞1�(�)�({�∈ℝ:�+1�(�)>�}) we show that for all �∈�+∞ we have the asymptotic estimate �+�(�)−1≲(1−�)1�[�]�+∞ for � sufficiently close to 1 and �>0 a numerical constant, and that this estimate is best possible. We also show that the reverse Hölder inequality for one-sided Muckenhoupt weights, previously proved by Martín-Reyes and de la Torre, is sharp, thus providing a quantitative equivalent definition of �+∞ . Our methods also allow us to show that a weight �∈�+∞ satisfies �∈�+� for all �>��[�]�+∞ .

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We are indebted to Francisco Javier Martín-Reyes for enlightening discussions related to the subject of the paper. The authors thank the referee for an expert reading and suggestions that helped improve the paper. P. Hagelstein: is partially supported by a grant from the Simons Foundation (#208831 to Paul Hagelstein). I. Parissis: is supported by Grant MTM2014-53850 of the Ministerio de Economía y Competitividad (Spain), Grant IT-641-13 of the Basque Government, and IKERBASQUE. O. Saari: is supported by the Academy of Finland and the Väisälä Foundation.

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