A note on generalized Poincaré-type inequalities with applications to weighted improved Poincaré-type inequalities
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The main result of this paper supports a conjecture by C. P\'erez and E. Rela about the properties of the weight appearing in their recent self-improving result of generalized inequalities of Poincar\'e-type in the Euclidean space. The result we obtain does not need any condition on the weight, but still is not fully satisfactory, even though the result by P\'erez and Rela is obtained as a corollary of ours. Also, we extend the conclusions of their theorem to the range $p<1$. As an application of our result, we give a unified vision of weighted improved Poincar\'e-type inequalities in the Euclidean setting, which gathers both weighted improved classical and fractional Poincar\'e inequalities within an approach which avoids any representation formula. We obtain results related to some already existing results in the literature and furthermore we improve them in some aspects. Finally, we also explore analog inequalities in the context of metric spaces by means of the already known self-improving results in this setting.