The weighted Stieltjes inequality and applications

Authors

  • Amiran Gogatishvili,

    Corresponding author
    1. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic
    • Phone: +420 222 090 786, Fax:+420 222 090 701
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  • Alois Kufner,

    Corresponding author
    1. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic
    • Phone: +420 222 090 741, Fax: +420 222 090 701
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  • Lars-Erik Persson

    Corresponding author
    1. Department of Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden
    2. Narvik University College, P. O. Box 385, N 8505 Narvik, Norway
    • Phone: +46 920 491 117, Fax:+46 920 491 073
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  • This paper is dedicated to Hans Triebel on the occasion of his 75th birthday

Abstract

Let 1 < pq < ∞. Inspired by some results concerning characterization of weighted Hardy type inequalities, where the equivalence of four scales of integral conditions was proved, we use related ideas to find some new equivalent scales of integral conditions related to the Stieltjes transform. By applying our result to weighted inequalities for the Stieltjes transform we obtain four new scales of conditions for characterization of this inequality. We also derive a new characterization for the solvability of a Riccati type equation and show via our new results that this characterization can be done in infinite many ways via our four scales of equivalent conditions.

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