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

We prove a local two-weight Poincaré inequality for cubes using the sparse domination method that has been influential in harmonic analysis. The proof involves a localized version of the Fefferman--Stein inequality for the sharp maximal function. By establishing a local-to-global result in a bounded domain satisfying a Boman chain condition, we show a two-weight $p$-Poincaré inequality in such domains. As an application we show that certain nonnegative supersolutions of the $p$-Laplace equation and distance weights are $p$-admissible in a bounded domain, in the sense that they support versions of the $p$-Poincaré inequality.

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