Premium
Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities on half‐spaces via mass transport and consequences
Author(s) -
Nguyen Van Hoang
Publication year - 2015
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/pdv026
Subject(s) - mathematics , nirenberg and matthaei experiment , sobolev space , inequality , sobolev inequality , pure mathematics , mathematical analysis
By adapting the mass transportation technique of Cordero‐Erausquin, Nazaret and Villani, we obtain a family of sharp Sobolev and Gagliardo–Nirenberg (GN) inequalities on the half‐spaceR n − 1 × R + , n ⩾ 1 equipped with the weight ω ( x ) = x n a , a ⩾ 0 . It amounts to work with the fractional dimensionn a = n + a . The extremal functions in the weighted Sobolev inequalities are fully characterized. Using a dimension reduction argument and the weighted Sobolev inequalities, we can reproduce a subfamily of the sharp GN inequalities on the Euclidean space due to Del Pino and Dolbeault, and obtain some new sharp GN inequalities as well. Our weighted inequalities are also extended to the domainR n − m × R + mand the weights are ω ( x , t ) = t 1 a 1 ⋯ t m a m, where n ⩾ m , m ⩾ 0 anda 1 , … , a m ⩾ 0 . A weighted L p ‐logarithmic Sobolev inequality is derived from these inequalities.