Poincaré inequality: From remote balls to all balls

• The uniform Poincare inequality for all balls is obtained using that of the Z-remote balls.
• The subset Z can separate the space into two or more connected components.
• The result can be applied to prove the Poincare inequality on weighted Dirichlet spaces — a simple example is also given.

It is shown that the Poincaré inequality holds uniformly for all balls whenever it holds for Z-remote balls providing the set Z satisfies some additional conditions including the condition that Z does not separate the space. The aim of this paper is to prove similar results without using this assumption.