On certain generalizations of the Gagliardo-Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities

摘要

We derive the inequality with a constant C(M, h) independent of f, where f belongs locally to the Sobolev space W~(2,1) (?) and f′ has compact support. Here M is an arbitrary N-function satisfying certain assumptions, h is a given function and τ_h(·) is its given transform independent of M. When M(λ) = λ~p and h ≡ 1 we retrieve the well-known inequality. We apply our inequality to obtain some generalizations of capacitary estimates and isoperimetric inequalities due to Maz'ya (1985).