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Published online by Cambridge University Press: 14 August 2020
In this expository article we give an overview of recent developments in the study of optimal Hardy-type inequalityin the continuum and in the discrete setting. In particular, we present the technique of the {\emph supersolution construction} that yield “as large as possibleȍ Hardy weightswhich is made precise in terms ofthe notion of criticality. Instead of presenting the most general setting possible, we restrict ourselves to the case of the Laplacian on smooth manifolds and bounded combinatorial graphs. Although the results hold in far greater generality, the fundamental phenomena as well as the core ideas of the proofs become especially clear in these basic settings.
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