Abstract
R. Coifman and C. Fefferman have shown for 1 < p < ∞ that the weighted Lp norm of the Hilbert transform is bounded by the weighted Lp norm of the Hardy-Littlewood maximal function if the weight function satisfies the condition A∞. It is shown in the first part of this paper that A∞ is not a necessary condition by deriving a large class of weight functions not in A∞ for which the norm inequality holds. The rest of the paper consists of the derivation of a necessary condition for the norm inequality; this condition closely resembles the A∞ condition.
Supported in part by N.S.F. Grant MCS 80-03098.
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© 1981 Birkhäuser Verlag Basel
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Muckenhoupt, B. (1981). Norm Inequalities Relating the Hilbert Transform to the Hardy-Littlewood Maximal Function. In: Butzer, P.L., Sz.-Nagy, B., Görlich, E. (eds) Functional Analysis and Approximation. ISNM 60: International Series of Numerical Mathematics / ISNM 60: Internationale Schriftenreihe zur Numerischen Mathematik / ISNM 60: Série internationale d’Analyse numérique, vol 60. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9369-5_21
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DOI: https://doi.org/10.1007/978-3-0348-9369-5_21
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