# Weighted ${L}_{\Phi}$ integral inequalities for operators of Hardy type

Studia Mathematica (1994)

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

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topBloom, 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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- [11] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991. Zbl0724.46032
- [12] E. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1-11. Zbl0508.42023
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