Sobolev inequalities with jointly concave weights on convex cones

Abstract

Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form ∫ E | u ( x ) | q ω ( x ) d x 1 / q ⩽ K 0 ∫ E | ∇ u ( x ) | p σ ( x ) d x 1 / p , u ∈ C 0 ∞ ( R n ) , (WSI)where p ⩾ 1 and q > 0 is the corresponding Sobolev critical exponent. Here E ⊆ R n is an open convex cone, and ω , σ : E → ( 0 , ∞ ) are two homogeneous weights verifying a general concavity-type structural condition. The constant K 0 = K 0 ( n , p , q , ω , σ ) > 0 is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that K 0 is optimal in (WSI) if and only if ω and σ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to partial differential equations are also provided.