Affine Moser–Trudinger and Morrey–Sobolev inequalities

  • Andrea Cianchi
  • Erwin Lutwak
  • Deane Yang
  • Gaoyong Zhang
Article

DOI: 10.1007/s00526-009-0235-4

Cite this article as:
Cianchi, A., Lutwak, E., Yang, D. et al. Calc. Var. (2009) 36: 419. doi:10.1007/s00526-009-0235-4

Abstract

An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard Ln energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the Ln Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard Lp energy, with p > n, is replaced by the affine energy.

Mathematics Subject Classification (2000)

46E35 46E30 

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Andrea Cianchi
    • 1
  • Erwin Lutwak
    • 2
  • Deane Yang
    • 2
  • Gaoyong Zhang
    • 2
  1. 1.Dipartimento di Matematica e Applicazioni per l’ArchitetturaUniversità di FirenzeFirenzeItaly
  2. 2.Department of MathematicsPolytechnic Institute of NYUBrooklynUSA

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