On a variant of the Hardy inequality between weighted Orlicz spaces

Agnieszka Ka/lamajska¹ ; Katarzyna Pietruska-Pa/luba¹

¹ Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Studia Mathematica, Tome 193 (2009) no. 1, pp. 1-28

Résumé

Let $M$ be an $N$-function satisfying the $\Delta_2$-condition, and let $\omega, \varphi$ be two other functions, with $\omega\ge 0$. We study Hardy-type inequalities $$ \int_{{\mathbb R}_+} M(\omega (x)|u(x)|) \exp (-\varphi (x))\,dx \le C\int_{{\mathbb R}_+} M(|u'(x)|) \exp (-\varphi (x))\,dx, $$ where $u$ belongs to some set ${\cal R }$ of locally absolutely continuous functions containing $C_0^\infty ({\mathbb R}_+)$. We give sufficient conditions on the triple $(\omega,\varphi,M)$ for such inequalities to be valid for all $u$ from a given set ${\cal R}$. The set ${\cal R}$ may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.

Détail

DOI : 10.4064/sm193-1-1

Keywords: n function satisfying delta condition omega varphi other functions omega study hardy type inequalities int mathbb omega exp varphi int mathbb exp varphi where belongs set cal locally absolutely continuous functions containing infty mathbb sufficient conditions triple omega varphi inequalities valid given set cal set cal may smaller set hardy transforms bounds constants given yielding classical hardy inequalities best constants