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

top
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

top

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

top
  1. [1] M. Artola, untitled and unpublished manuscript. 
  2. [2] S. Bloom and R. Kerman, Weighted norm inequalities for operators of Hardy type, Proc. Amer. Math. Soc. 113 (1991), 135-141. Zbl0753.42010
  3. [3] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408. Zbl0402.26006
  4. [4] H. Heinig and L. Maligranda, Interpolation with weights in Orlicz spaces, Luleå University of Technology, Department of Applied Mathematics Research Report 03 (1992). Zbl0821.46024
  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). 
  6. [6] R. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1981/82), 277-284. Zbl0517.42030
  7. [7] M. A. Krasnosel'skii [M. A. Krasnosel'skiĭ] and Ya. B. Rutickii [Ya. B. Rutitskiĭ], Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961. Zbl0095.09103
  8. [8] F. Martín-Reyes and E. Sawyer, Weighted norm inequalities for the Riemann-Liouville fractional integral operators, Proc. Amer. Math. Soc. 106 (1989), 727-733. Zbl0704.42018
  9. [9] B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 34 (1972), 31-38. Zbl0236.26015
  10. [10] L. Quinsheng, Two weight Φ-inequalities for the Hardy operator, Hardy-Littlewood maximal operator and fractional integrals, Proc. Amer. Math. Soc., to appear. Zbl0758.42012
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.