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Note on the concentration-compactness principle for generalized Moser-Trudinger inequalities

Robert Černý (2012)

Open Mathematics

Let Ω ⊂ ℝn, n ≥ 2, be a bounded domain and let α < n − 1. Motivated by Theorem I.6 and Remark I.18 of [Lions P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1985, 1(1), 145–201] and by the results of [Černý R., Cianchi A., Hencl S., Concentration-Compactness Principle for Moser-Trudinger inequalities: new results and proofs, Ann. Mat. Pura Appl. (in press), DOI: 10.1007/s10231-011-0220-3], we give a sharp estimate of the...

On a variant of the Hardy inequality between weighted Orlicz spaces

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba (2009)

Studia Mathematica

Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities ∫ ℝ ₊ M ( ω ( x ) | u ( x ) | ) e x p ( - φ ( x ) ) d x ≤ C ∫ ℝ ₊ M ( | u ' ( x ) | ) e x p ( - φ ( x ) ) d x , where u belongs to some set of locally absolutely continuous functions containing C ₀ ∞ ( ℝ ₊ ) . We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set . The set may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.

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