New approach to the affine Pólya-Szegö principle and the stability version of the affine Sobolev inequality

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• 作者：Van Hoang Nguyen ^{vanhoang0610@yahoo.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：26D07 ; 46E35 ; 52A40
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：22 October 2016
• 年：2016
• 卷：302
• 期：Complete
• 页码：1080-1110
• 全文大小：520 K

文摘

Inspired by a recent work of Haddad, Jiménez and Montenegro, we give a new and simple approach to the recently established general affine Pólya–Szegö principle. Our approach is based on the general L_p$L_p$ Busemann–Petty centroid inequality and does not rely on the general L_p$L_p$ Petty projection inequality or the solution of the L_p$L_p$ Minkowski problem. A Brothers–Ziemer-type result for the general affine Pólya–Szegö principle is also established. As applications, we reprove some sharp affine Sobolev-type inequalities and settle their equality conditions. We also prove a stability estimate for the affine Sobolev inequality on functions of bounded variation by using our new approach. As a corollary of this stability result, we deduce a stability estimate for the affine logarithmic-Sobolev inequality.