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Sobolev inequalities with jointly concave weights on convex cones
Author(s) -
Balogh Zoltán M.,
Gutiérrez Cristian E.,
Kristály Alexandru
Publication year - 2021
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms.12384
Subject(s) - mathematics , sobolev space , combinatorics , regular polygon , sobolev inequality , type (biology) , cone (formal languages) , multiplicative function , exponent , convex body , mathematical analysis , pure mathematics , discrete mathematics , convex optimization , geometry , ecology , linguistics , philosophy , algorithm , biology
Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form WSI∫ E| u ( x ) | qω ( x )d x 1 / q ⩽ K 0∫ E| ∇ u ( x ) | pσ ( x )d x 1 / p ,u ∈ C 0 ∞ ( R n ) , where p ⩾ 1 and q > 0 is the corresponding Sobolev critical exponent. Here E ⊆ R nis an open convex cone, and ω , σ : E → ( 0 , ∞ ) are two homogeneous weights verifying a general concavity‐type structural condition. The constantK 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.