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Affine Moser–Trudinger and Morrey–Sobolev inequalities

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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 L n 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 L n Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L p energy, with p > n, is replaced by the affine energy.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni per l’Architettura, Università di Firenze, Piazza Ghiberti 27, 50122, Firenze, Italy

    Andrea Cianchi

  2. Department of Mathematics, Polytechnic Institute of NYU, Brooklyn, New York, USA

    Erwin Lutwak, Deane Yang & Gaoyong Zhang

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  1. Andrea Cianchi
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  2. Erwin Lutwak
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Correspondence to Andrea Cianchi.

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Cianchi, A., Lutwak, E., Yang, D. et al. Affine Moser–Trudinger and Morrey–Sobolev inequalities. Calc. Var. 36, 419–436 (2009). https://doi.org/10.1007/s00526-009-0235-4

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