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Sharp constants for Moser-type inequalities concerning embeddings into Zygmund spaces

Robert Černý (2012)

Commentationes Mathematicae Universitatis Carolinae

Let n ≥ 2 and Ω ⊂ ℝ n be a bounded set. We give a Moser-type inequality for an embedding of the Orlicz-Sobolev space W 0 L Φ ( Ω ) , where the Young function Φ behaves like t n log α ( t ) , α < n - 1 , for t large, into the Zygmund space Z 0 n - 1 - α n ( Ω ) . We also study the same problem for the embedding of the generalized Lorentz-Sobolev space W 0 m L n m , q log α L ( Ω ) , m < n , q ∈ ( 1 , ∞ ] , α < 1 q ' , embedded into the Zygmund space Z 0 1 q ' - α ( Ω ) .

Shift inequalities of Gaussian type and norms of barycentres

F. Barthe, D. Cordero-Erausquin, M. Fradelizi (2001)

Studia Mathematica

We derive the equivalence of different forms of Gaussian type shift inequalities. This completes previous results by Bobkov. Our argument strongly relies on the Gaussian model for which we give a geometric approach in terms of norms of barycentres. Similar inequalities hold in the discrete setting; they improve the known results on the so-called isodiametral problem for the discrete cube. The study of norms of barycentres for subsets of convex bodies completes the exposition.

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