Self-Improving Properties of John–Nirenberg and Poincaré Inequalities on Spaces of Homogeneous Type

We give a condition which ensures that if one inequality of Sobolev–Poincaré type is valid then other stronger inequalities of a similar type also hold, including weighted versions. Our main result includes many previously known results as special cases. We carry out the analysis in the context of spaces of homogeneous type, but the main result is new even in the usual Euclidean setting.