A Carlson type inequality with blocks and interpolation

Natan Kruglyak; Lech Maligranda; Lars Persson

Studia Mathematica (1993)

  • Volume: 104, Issue: 2, page 161-180
  • ISSN: 0039-3223

Abstract

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An inequality, which generalizes and unifies some recently proved Carlson type inequalities, is proved. The inequality contains a certain number of “blocks” and it is shown that these blocks are, in a sense, optimal and cannot be removed or essentially changed. The proof is based on a special equivalent representation of a concave function (see [6, pp. 320-325]). Our Carlson type inequality is used to characterize Peetre’s interpolation functor ⟨ ⟩ φ (see [26]) and its Gagliardo closure on couples of functional Banach lattices in terms of the Calderón-Lozanovskiǐ construction. Our interest in this functor is inspired by the fact that if φ = t θ ( 0 < θ < 1 ) , then, on couples of Banach lattices and their retracts, it coincides with the complex method (see [20], [27]) and, thus, it may be regarded as a “real version” of the complex method.

How to cite

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Kruglyak, Natan, Maligranda, Lech, and Persson, Lars. "A Carlson type inequality with blocks and interpolation." Studia Mathematica 104.2 (1993): 161-180. <http://eudml.org/doc/215967>.

@article{Kruglyak1993,
abstract = {An inequality, which generalizes and unifies some recently proved Carlson type inequalities, is proved. The inequality contains a certain number of “blocks” and it is shown that these blocks are, in a sense, optimal and cannot be removed or essentially changed. The proof is based on a special equivalent representation of a concave function (see [6, pp. 320-325]). Our Carlson type inequality is used to characterize Peetre’s interpolation functor $⟨⟩_\{φ\}$ (see [26]) and its Gagliardo closure on couples of functional Banach lattices in terms of the Calderón-Lozanovskiǐ construction. Our interest in this functor is inspired by the fact that if $φ = t^\{θ\}(0 < θ < 1)$, then, on couples of Banach lattices and their retracts, it coincides with the complex method (see [20], [27]) and, thus, it may be regarded as a “real version” of the complex method.},
author = {Kruglyak, Natan, Maligranda, Lech, Persson, Lars},
journal = {Studia Mathematica},
keywords = {concavity; Carlson's inequality; blocks; interpolation; Peetre's interpolation functor; Calderón-Lozanovskiǐ construction; Carlson type inequalities; Gagliardo closure on couples of functional Banach lattices; Calderón-Lozanovskij construction; couples of Banach lattices and their retracts; complex method},
language = {eng},
number = {2},
pages = {161-180},
title = {A Carlson type inequality with blocks and interpolation},
url = {http://eudml.org/doc/215967},
volume = {104},
year = {1993},
}

TY - JOUR
AU - Kruglyak, Natan
AU - Maligranda, Lech
AU - Persson, Lars
TI - A Carlson type inequality with blocks and interpolation
JO - Studia Mathematica
PY - 1993
VL - 104
IS - 2
SP - 161
EP - 180
AB - An inequality, which generalizes and unifies some recently proved Carlson type inequalities, is proved. The inequality contains a certain number of “blocks” and it is shown that these blocks are, in a sense, optimal and cannot be removed or essentially changed. The proof is based on a special equivalent representation of a concave function (see [6, pp. 320-325]). Our Carlson type inequality is used to characterize Peetre’s interpolation functor $⟨⟩_{φ}$ (see [26]) and its Gagliardo closure on couples of functional Banach lattices in terms of the Calderón-Lozanovskiǐ construction. Our interest in this functor is inspired by the fact that if $φ = t^{θ}(0 < θ < 1)$, then, on couples of Banach lattices and their retracts, it coincides with the complex method (see [20], [27]) and, thus, it may be regarded as a “real version” of the complex method.
LA - eng
KW - concavity; Carlson's inequality; blocks; interpolation; Peetre's interpolation functor; Calderón-Lozanovskiǐ construction; Carlson type inequalities; Gagliardo closure on couples of functional Banach lattices; Calderón-Lozanovskij construction; couples of Banach lattices and their retracts; complex method
UR - http://eudml.org/doc/215967
ER -

References

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