On a variant of Hardy inequality between weighted Orlicz spaces

Abstract

Let M be an N-function satisfying the ${\Delta }_2$- condition, let $ω, \vp$ be two other functions, $\omega \ge 0$. 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 $\mathcal{R}$ 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 $\mathcal{R}$. The set $\mathcal{R}$ can be smaller than the set of Hardy transforms. Bounds for constants, retrieving classical Hardy inequalities with best constants, are also given.

34 pages

Cited by in corpus (1)

• On a variant of Gagliardo-Nirenberg inequality deduced from Hardy