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

Abstrakt

Poincaré-Sobolev-type inequalities involving rearrangement-invariant norms on the entire Rn are provided. Namely, inequalities of the type ‖u-P‖Y(Rn)≤C‖∇mu‖X(Rn), where X and Y are either rearrangement-invariant spaces over Rn or Orlicz spaces over Rn, u is a m- times weakly differentiable function whose gradient is in X, P is a polynomial of order at most m- 1 , depending on u, and C is a constant independent of u, are studied.

In a sense optimal rearrangement-invariant spaces or Orlicz spaces Y in these inequalities when the space X is fixed are found. A variety of particular examples for customary function spaces are also provided.

Klíčová slova

• Optimal spaces
• Poincaré inequality
• Poincaré-Sobolev inequality
• Rearrangement-invariant spaces
• Sobolev spaces