On a variant of Hardy inequality between weighted Orlicz spaces

Kalamajska, Agnieszka; Pietruska-Paluba, Katarzyna

On a variant of Hardy inequality between weighted Orlicz spaces

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Let M be an N-function satisfying the $Δ_2$- condition, let $ω, \vp$ be two other functions, $ω\ge 0$. We study Hardy-type inequalities \[ \int_{\rp} M(ω(x)|u(x)|) {\rm exp}(-\vp (x))dx \le C\int_{\rp} M(|u'(x)|) {\rm exp}(-\vp (x))dx, \] where $u$ belongs to some dilation invariant set ${\cal R}$ contained in the space of locally absolutely continuous functions. We give sufficient conditions the triple $(ω,\vp,M)$ must satisfy in order to have such inequalities valid for $u$ from a given set ${\cal R}$. The set ${\cal R}$ can be smaller than the set of Hardy transforms. Bounds for constants, retrieving classical Hardy inequalities with best constants, are also given.
34 pages