On a variant of Hardy inequality between weighted Orlicz spaces

TL;DR

This paper investigates a variant of Hardy inequalities within weighted Orlicz spaces, providing conditions for their validity, bounds for constants, and connections to classical inequalities.

Contribution

It introduces new sufficient conditions for Hardy-type inequalities in weighted Orlicz spaces and derives bounds for the associated constants.

Findings

Established conditions for Hardy inequalities in weighted Orlicz spaces.

Derived bounds for the constants involved in the inequalities.

Connected the results to classical Hardy inequalities with optimal constants.

Abstract

Let M be an N-function satisfying the $\Delta_2$- condition, let $\omega, \vp$ be two other functions, $\omega\ge 0$. We study Hardy-type inequalities \[ \int_{\rp} M(\omega (x)|u(x)|) {\rm exp}(-\vp (x))dx \le C\int_{\rp} M(|u'(x)|) {\rm exp}(-\vp (x))dx, \] 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 $(\omega,\vp,M)$ 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.