New approach to the affine P 'olya-Szeg "o principle and the stability\n version of the affine Sobolev inequality

2015/06/24 by Van Hoang Nguyen, Nguyen, Van Hoang · 4 citations
Engineering · #26D07 #46E35 #52A40 #FOS: Mathematics #Fatigue and fracture mechanics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1506.07335

openalex publication_date 2015/06/24 · openalex created_date 2022/08/16 · openalex updated_date 2026/07/28

Abstract

Inspired by a recent work of Haddad, Jim 'enez and Montenegro, we give a new\nand simple approach to the recently established general affine P 'olya-Szeg "o\nprinciple. Our approach is based on the general L_p Busemann-Petty centroid\ninequality and does not rely on the general L_p Petty projection inequality\nor the solution of the L_p Minkowski problem. A Brothers-Ziemer-type result\nfor the general affine P 'olya-Szeg "o principle is also established. As\napplications, we reprove some sharp affine Sobolev-type inequalities and settle\ntheir equality conditions. We also prove a stability estimate for the affine\nSobolev inequality on functions of bounded variation by using our new approach.\nAs a corollary of this stability result, we deduce a stability estimate for the\naffine logarithmic--Sobolev inequality.\n

Cited by

• Affine isoperimetric inequalities for the first eigenvalue of the m-th order Affine p-Laplace Operator
• Sharp anisotropic L²-Caffarelli-Kohn-Nirenberg inequalities associated with the Minkowski functional
• Generalized affine Hardy-Littlewood-Sobolev inequalities
• Affine fractional Lp Polya-Szego inequalities

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