A Weighted Trudinger–Moser Inequality on R^N and Its Application to Grushin Operator  

作　　者：Jia Jun WANG Qiao Hua YANG 

机构地区：[1]School of Mathematics and Statistics,Wuhan University,Wuhan 430072,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2020年第4期363-378,共16页数学学报（英文版）

基　　金：Supported by the National Natural Science Foundation of China(Grant No.11201346)。

摘　　要：Let x=(x',x'')with x'∈Rk and x''∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x'-symmetric function u∈C0^∞(Ω)with∫Ω|■u(x)|^N-a|x'|^-adx≤1,the following uniform inequality holds1/∫Ω|x|^-adx∫Ωe^βa|u|N-a/N-a-1|x'|^-adx≤C,whereβa=(N-a)(2πN/2Γ(k-a/2)Γ(k/2)/Γ(k/2)r(N-a/2))1/N-a-1.Furthermore,βa can not be replaced by any greater number.As the application,we obtain some weighted Trudinger–Moser inequalities for x-symmetric function on Grushin space.

关 键 词：Trudinger–Moser inequality Grushin operator sharp constant H-type group 

分 类 号：O174[理学—数学] O178[理学—基础数学]