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Hardy inequalities and Assouad dimensions

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Abstract

We establish both sufficient and necessary conditions for weighted Hardy inequalities in metric spaces in terms of Assouad (co)dimensions. Our sufficient conditions in the case where the complement is thin are new, even in euclidean spaces, while in the case of a thick complement, we give new formulations for previously known sufficient conditions which reveal a natural duality between these two cases. Our necessary conditions are rather straight-forward generalizations from the unweighted case but, together with some examples, indicate the essential sharpness of our results. In addition, we consider the mixed case in which the complement may contain both thick and thin parts.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), FI-40014, Jyvaskyla, Finland

    Juha Lehrbäck

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  1. Juha Lehrbäck
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Correspondence to Juha Lehrbäck.

Additional information

The author has been supported by the Academy of Finland, grant no. 252108.

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Lehrbäck, J. Hardy inequalities and Assouad dimensions. JAMA 131, 367–398 (2017). https://doi.org/10.1007/s11854-017-0013-8

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  • DOI: https://doi.org/10.1007/s11854-017-0013-8

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