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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 M ( ω ( x ) | u ( x ) | ) e x p ( - φ ( x ) ) d x C M ( | u ' ( x ) | ) e x p ( - φ ( x ) ) d x , where u belongs to some set of locally absolutely continuous functions containing C ( ) . 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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