Poincaré-Sobolev inequalities with rearrangement-invariant norms on the entire space

Poincaré-Sobolev-type inequalities involving rearrangement-invariant norms on the entire $$\mathbb R^n$$ are provided. Namely, of type $$\Vert u-P\Vert _{Y(\mathbb R^n)}\le C\Vert \nabla ^m u\Vert _{X(\mathbb R^n)}$$ , where X and Y either spaces over or Orlicz u is a $$m-$$ times weakly differentiable function whose gradient in X, P polynomial order at most $$m-1$$ depending u, C constant independent studied. In sense optimal these when space fixed found. A variety particular examples for customary also