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We prove a local two-weight Poincar\'e inequality for cubes using the sparsedomination method that has been influential in harmonic analysis. The proofinvolves a localized version of the Fefferman--Stein inequality for the sharpmaximal function. By establishing a local-to-global result in a bounded domainsatisfying a Boman chain condition, we show a two-weight $p$-Poincar\'einequality in such domains. As an application we show that certain nonnegativesupersolutions of the $p$-Laplace equation and distance weights are$p$-admissible in a bounded domain, in the sense that they support versions ofthe $p$-Poincar\'e inequality.

Weighted norm inequalities in a bounded domain by the sparse domination method