Sobolev inequality with non-uniformly degenerating gradient

Abstract

In this paper we prove the following weighted Sobolev inequality in a bounded domain Ω ⊂ R n , n ≥ 1 , of a homogeneous space ( R n , ρ , w d x ) , under suitable compatibility conditions on the positive weight functions ( v , w , ω 1 , ω 2 , … , ω n ) and on the quasi-metric ρ , ( ∫ Ω | f | q v w d z ) 1 q ≤ C ∑ i = 1 N ( ∫ Ω | f z i | p ω i M S w d z ) 1 p , f ∈ L i p 0 ( Ω ¯ ) , where q ≥ p > 1 and M S denotes the strong maximal operator. Some corollaries on non-uniformly degenerating gradient inequalities are derived.