On a variant of the Hardy inequality between weighted Orlicz spaces

Tom 193 / 2009

Agnieszka Ka/lamajska, Katarzyna Pietruska-Pa/luba Studia Mathematica 193 (2009), 1-28 MSC: Primary 26D10; Secondary 46E35. DOI: 10.4064/sm193-1-1

Streszczenie

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.

Autorzy

  • Agnieszka Ka/lamajskaInstitute of Mathematics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
  • Katarzyna Pietruska-Pa/lubaInstitute of Mathematics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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