A study of some constants characterizing the weighted Hardy inequality

Volume 64 / 2004

Alois Kufner, Lars-Erik Persson, Anna Wedestig Banach Center Publications 64 (2004), 135-146 MSC: Primary 26D10, 26D15; Secondary 47B07, 47B38. DOI: 10.4064/bc64-0-11

Abstract

The modern form of Hardy's inequality means that we have a necessary and sufficient condition on the weights $u$ and $v$ on $[0,b]$ so that the mapping $$ H:L^{p}(0,b;v)\rightarrow L^{q}(0,b;u) $$ is continuous, where $Hf(x)=\int_{0}^{x}f(t)dt$ is the Hardy operator. We consider the case $1< p\leq q< \infty $ and then this condition is usually written in the Muckenhoupt form $$ A_{1}:=\sup _{0< x< b}A_{M}(x)< \infty . \tag*{$(*)$}$$ In this paper we discuss and compare some old and new other constants $A_{i}$ of the form $(*)$, which also characterize Hardy's inequality. We also point out some dual forms of these characterizations, prove some new compactness results and state some open problems.

Authors

  • Alois KufnerMatematical Institute
    Academy of Sciences of the Czech Republic
    Žitná 25, 115 67 Praha 1
    Czech Republic
    e-mail
  • Lars-Erik PerssonDepartment of Mathematics
    Luleå University of Technology
    SE-971 87 Luleå, Sweden
    e-mail
  • Anna WedestigDepartment of Mathematics
    Luleå University of Technology
    SE-971 87 Luleå, Sweden
    e-mail

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