Construction of an optimal envelope for a cone of nonnegative functions with monotonicity properties

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Abstract

We study the problem of constructing a minimal quasi-Banach ideal space containing a given cone of nonnegative functions with monotonicity properties. The construction employs nondegenerate operators. We present general results on constructing optimal envelopes consistent with an order relation and obtain specifications of these constructions for various cones and various order relations. We also address the issue of order covering and order equivalence of cones.

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References

  1. 1.
    T. Aoki, “Locally bounded linear topological spaces,” Proc. Imp. Acad. Tokyo 18, 588–594 (1942).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    E. G. Bakhtigareeva and M. L. Goldman, “Associated norms and optimal embeddings for a class of two-weight integral quasinorms,” Fundam. Prikl. Mat. 19 (5), 3–33 (2014).MathSciNetGoogle Scholar
  3. 3.
    E. G. Bakhtigareeva, M. L. Goldman, and P. P. Zabreiko, “Optimal reconstruction of a generalized Banach function space from a cone of nonnegative functions,” Vestn. Tambov. Univ., Ser. Estestv. Tekh. Nauki 19 (2), 316–330 (2014).Google Scholar
  4. 4.
    C. Bennett and R. Sharpley, Interpolation of Operators (Academic, New York, 1988).MATHGoogle Scholar
  5. 5.
    J. Bergh and J. Löfström, Interpolation Spaces. An Introduction (Springer, Berlin, 1976).CrossRefMATHGoogle Scholar
  6. 6.
    L. Diening and S. Samko, “Hardy inequality in variable exponent Lebesgue spaces,” Fract. Calc. Appl. Anal. 10 (1), 1–18 (2007).MathSciNetMATHGoogle Scholar
  7. 7.
    A. Gogatishvili and V. D. Stepanov, “Reduction theorems for weighted integral inequalities on the cone of monotone functions,” Usp. Mat. Nauk 68 (4), 3–68 (2013) [Russ. Math. Surv. 68, 597–664 (2013)].MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    M. L. Goldman, “Optimal embeddings of generalized Bessel and Riesz potentials,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 269, 91–111 (2010) [Proc. Steklov Inst. Math. 269, 85–105 (2010)].MathSciNetMATHGoogle Scholar
  9. 9.
    M. L. Goldman, “Some constructive criteria of optimal embeddings for potentials,” Complex Var. Elliptic Eqns. 56 (10–11), 885–903 (2011).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    M. L. Goldman and D. D. Haroske, “Estimates for continuity envelopes and approximation numbers of Bessel potentials,” J. Approx. Theory 172, 58–85 (2013).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    M. L. Goldman and D. Haroske, “Optimal Calderon spaces for generalized Bessel potentials,” Dokl. Akad. Nauk 463 (1), 14–17 (2015) [Dokl. Math. 92 (1), 404–407 (2015)].MathSciNetMATHGoogle Scholar
  12. 12.
    M. L. Goldman and P. P. Zabreiko, “Optimal reconstruction of a Banach function space from a cone of nonnegative functions,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 284, 142–156 (2014) [Proc. Steklov Inst. Math. 284, 133–147 (2014)].MathSciNetMATHGoogle Scholar
  13. 13.
    S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators (Nauka, Moscow, 1978; Am. Math. Soc., Providence, RI, 1982), Transl. Math. Monogr. 54.Google Scholar
  14. 14.
    K. Leśnik and L. Maligranda, “Abstract Cesàro spaces. Duality,” J. Math. Anal. Appl. 424 (2), 932–951 (2015).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    K. Leśnik and L. Maligranda, “Abstract Cesàro spaces. Optimal range,” Integral Eqns. Oper. Theory 81 (2), 227–235 (2015).CrossRefMATHGoogle Scholar
  16. 16.
    S. Rolewicz, “On a certain class of linear metric spaces,” Bull. Acad. Pol. Sci., Cl. III, 5, 471–473 (1957).MathSciNetMATHGoogle Scholar
  17. 17.
    V. D. Stepanov, “On optimal Banach spaces containing a weighted cone of monotone or quasi-concave functions,” Dokl. Akad. Nauk 464 (2), 145–147 (2015) [Dokl. Math. 92 (2), 545–547 (2015)].MathSciNetMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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