On F-Sobolev and Orlicz-Sobolev inequalities

Abstract

Let F ∈ C([0,∞)) be a positive increasing function such that Φ(s) := sF(s) is a Young function. In general, the F-Sobolev inequality and the Φ-Orlicz-Sobolev inequality are not equivalent. In this paper, a growth condition on F is presented for these two inequalities to be equivalent. The main result generalizes the corresponding known one for F(s) = log^δ(1+s) (δ>0). As an application, some criteria are presented for the F-Sobolev inequality to hold.

Keywords

Orlicz-Sobolev inequality / F-Sobolev inequality / super Poincar&#233 / inequality