Sharp inequalities for one-sided Muckenhoupt weights

• Autores: Paul Alton Hagelstein , Ioannis Parissis , Olli Saari
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 69, Fasc. 1, 2018, págs. 151-161
• Idioma: inglés
• DOI: 10.1007/s13348-017-0201-y
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• Resumen

  • 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?}) 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 ?>??[?]?+∞ .