Weighted 1-dimensional Orlicz-Poincar\'e inequalities

In this paper we establish necessary and sufficient conditions for weighted Orlicz-Poincar\'e inequalities in dimension one. Our theorems generalize the main results of Chua and Wheeden, who established necessary and sufficient conditions for weighted $(q,p)$ Poincar\'e inequalities. We give an example of a weight satisfying sufficient conditions for a $(\Phi, p)$ Orlicz-Poincar\'e inequality where the gauge norm with respect to $\Phi$ is a bump on the Lebesgue $L^p$ norm. This weight, on the other hand, does not satisfy a $(q,p)$ Poincar\'e inequality for any $q > p$.