Weighted 1-dimensional Orlicz-Poincaré inequalities

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