Weighted L Φ integral inequalities for operators of Hardy type

Steven Bloom; Ron Kerman

Studia Mathematica (1994)

  • Volume: 110, Issue: 1, page 35-52
  • ISSN: 0039-3223

Abstract

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Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for Φ 2 - 1 ( ʃ Φ 2 ( w ( x ) | T f ( x ) | ) t ( x ) d x ) Φ 1 - 1 ( ʃ Φ 1 ( C u ( x ) | f ( x ) | ) v ( x ) d x ) to hold when Φ 1 and Φ 2 are N-functions with Φ 2 Φ 1 - 1 convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.

How to cite

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Bloom, Steven, and Kerman, Ron. "Weighted $L_{Φ}$ integral inequalities for operators of Hardy type." Studia Mathematica 110.1 (1994): 35-52. <http://eudml.org/doc/216097>.

@article{Bloom1994,
abstract = {Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for $Φ_2^\{-1\} (ʃΦ_2(w(x)|Tf(x)|)t(x)dx) ≤ Φ_\{1\}^\{-1\}(ʃΦ_\{1\}(Cu(x)|f(x)|)v(x)dx)$ to hold when $Φ_1$ and $Φ_2$ are N-functions with $Φ_2∘Φ_\{1\}^\{-1\}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.},
author = {Bloom, Steven, Kerman, Ron},
journal = {Studia Mathematica},
keywords = {weighted inequalities; integral operators of Hardy type; Orlicz spaces; norm inequality; outer modular inequality; inner modular inequality; Sawyer inequality; four-weight inequality},
language = {eng},
number = {1},
pages = {35-52},
title = {Weighted $L_\{Φ\}$ integral inequalities for operators of Hardy type},
url = {http://eudml.org/doc/216097},
volume = {110},
year = {1994},
}

TY - JOUR
AU - Bloom, Steven
AU - Kerman, Ron
TI - Weighted $L_{Φ}$ integral inequalities for operators of Hardy type
JO - Studia Mathematica
PY - 1994
VL - 110
IS - 1
SP - 35
EP - 52
AB - Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for $Φ_2^{-1} (ʃΦ_2(w(x)|Tf(x)|)t(x)dx) ≤ Φ_{1}^{-1}(ʃΦ_{1}(Cu(x)|f(x)|)v(x)dx)$ to hold when $Φ_1$ and $Φ_2$ are N-functions with $Φ_2∘Φ_{1}^{-1}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.
LA - eng
KW - weighted inequalities; integral operators of Hardy type; Orlicz spaces; norm inequality; outer modular inequality; inner modular inequality; Sawyer inequality; four-weight inequality
UR - http://eudml.org/doc/216097
ER -

References

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  1. [1] M. Artola, untitled and unpublished manuscript. 
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  3. [3] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408. Zbl0402.26006
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  5. [5] S. S. Kazarian, Integral inequalities in Orlicz reflexive weighted spaces for the conjugate function, Dokl. Akad. Nauk Armyan. SSR 25 (3) (1990), 261-273 (in Russian). 
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  11. [11] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991. Zbl0724.46032
  12. [12] E. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1-11. Zbl0508.42023
  13. [13] E. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 302 (1988), 533-545. Zbl0665.42023
  14. [14] V. Stepanov, Two-weighted estimates for Riemann-Liouville integrals, preprint no. 39, Czech. Acad. Sci., 1988. 
  15. [15] G. Talenti, Osservazioni sopra una classe di disuguaglianze, Rend. Sem. Mat. Fis. Milano 39 (1969), 171-185. Zbl0218.26011
  16. [16] G. Tomaselli, A class of inequalities, Boll. Un. Mat. Ital. 21 (1969), 622-631. Zbl0188.12103
  17. [17] A. Zygmund, Trigonometric Series, Vol. I, 2nd ed., Cambridge Univ. Press, Cambridge, 1959. Zbl0085.05601

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