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Orlicz-Poincaré inequalities, maximal functions and AΦ-conditions

Published online by Cambridge University Press:  04 February 2010

Jana Björn
Affiliation:
Department of Mathematics, Linköpings Universitet, 58183 Linköping, Sweden, ([email protected])

Abstract

For a measure μ on ℝn (or on a doubling metric measure space) and a Young function Φ, we define two versions of Orlicz–Poincaré inequalities as generalizations of the usual p-Poincaré inequality. It is shown that, on ℝ, one of them is equivalent to the boundedness of the Hardy–Littlewood maximal operator from LΦ(ℝ,μ) to LΦ(ℝ,μ), while the other is equivalent to a generalization of the Muckenhoupt Ap-condition. While one direction in these equivalences is valid only on ℝ, the other holds in the general setting of doubling metric measure spaces. We also characterize both Orlicz–Poincaré inequalities on metric measure spaces by means of pointwise inequalities involving maximal functions of the gradient.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2010

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