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We prove a general self-improvement property for a family of weightedpointwise inequalities on open sets, including pointwise Hardy inequalitieswith distance weights. For this purpose we introduce and study the classes of$p$-Poincar\'e and $p$-Hardy weights for an open set $\Omega\subset X$, where$X$ is a metric measure space. We also apply the self-improvement of weightedpointwise Hardy inequalities in connection with usual integral versions ofHardy inequalities.

Self-improvement of weighted pointwise inequalities on open sets