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On equivalence of super log Sobolev and Nash type inequalities

Marco Biroli, Patrick Maheux (2014)

Colloquium Mathematicae

We prove the equivalence of Nash type and super log Sobolev inequalities for Dirichlet forms. We also show that both inequalities are equivalent to Orlicz-Sobolev type inequalities. No ultracontractivity of the semigroup is assumed. It is known that there is no equivalence between super log Sobolev or Nash type inequalities and ultracontractivity. We discuss Davies-Simon's counterexample as the borderline case of this equivalence and related open problems.

On generalized Moser-Trudinger inequalities without boundary condition

Robert Černý (2012)

Czechoslovak Mathematical Journal

We give a version of the Moser-Trudinger inequality without boundary condition for Orlicz-Sobolev spaces embedded into exponential and multiple exponential spaces. We also derive the Concentration-Compactness Alternative for this inequality. As an application of our Concentration-Compactness Alternative we prove that a functional with the sub-critical growth attains its maximum.

On Hardy q -inequalities

Lech Maligranda, Ryskul Oinarov, Lars-Erik Persson (2014)

Czechoslovak Mathematical Journal

Some q -analysis variants of Hardy type inequalities of the form ∫ 0 b x α - 1 ∫ 0 x t - α f ( t ) d q t p d q x ≤ C ∫ 0 b f p ( t ) d q t with sharp constant C are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.

On overdetermined Hardy inequalities

Alois Kufner, Herbert Leinfelder (1998)

Mathematica Bohemica

Necessary and sufficient condition on the weights will be derived under which a k -th order Hardy inequality holds on classes of functions satisfying more than k “boundary” conditions.

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