JIPAM ] Up ]

 


Volume 1, Issue 1, 2000

Article 10

http://jipam.vu.edu.au/v1n1/010_99.html

WEIGHTED MODULAR INEQUALITIES FOR HARDY-TYPE OPERATORS ON MONOTONE FUNCTIONS

HANS P. HEINIG AND QINSHENG LAI
E-Mail: heinig@mcmail.cis.mcmaster.ca,
jlai@comnetix.com

DEPARTMENT OF MATHEMATICS AND STATISTICS, McMASTER UNIVERSITY, HAMILTON, ONTARIO, L8S 4K1, CANADA.

Received 3 November, 1999; accepted 31 January, 2000.
Communicated by: B. Opic.


If

\begin{displaymath}(Kf)(x)=\int_{0}^{x}k(x,y)f(y)\,dy,\end{displaymath}

$x>0$, is a Hardy-type operator defined on the cone of monotone functions, then weight characterizations for which the modular inequality

\begin{displaymath}
Q^{-1}\left( \int_{0}^{\infty }Q[\theta (Kf)]w\right) \leq P^{-1}\left(
\int_{0}^{\infty }P[Cf]v\right)
\end{displaymath}

holds, are given for a large class of modular functions $P,Q$. Specifically, these functions need not both be $N$-functions, and the class includes the case where $Q\circ P^{-1}$ is concave. Our results generalize those in [7,24], where the case $Q\circ P^{-1}$ convex, with $P,Q$, $N$-function was studied. Applications involving the Hardy averaging operator, its dual, the Hardy-Littlewood maximal function, and the Hilbert transform are also given.

[7] P. DRÁBEK, H.P. HEINIG AND A. KUFNER, Weighted modular inequalities for monotone functions, J. of Inequal. and Appl., 1 (1997), 183–197.

[24] J.Q. SUN, The modular inequalities for a class of convolutions operators on monotone functions, Proc. Amer. Math. Soc., 125 (1997), 2293–2305.


Key words:
Hardy-type operators, modular inequalities, weights, N-functions, characterizations, Orlicz-Lorentz.

2000 Mathematics Subject Classification:
26D15, 42B25, 26A33, 46E30.


Download this article (PDF):

Suitable for a printer:       

Suitable for a monitor:        

To view these files we recommend you save them to your file system and then view by using the Adobe Acrobat Reader. 

That is, click on the icon using the 2nd mouse button and select "Save Target As..." (Microsoft Internet Explorer) or "Save Link As..." (Netscape Navigator).

See our PDF pages for more information.

 

 

Other papers in this issue

Volume 1, Number 1, 2000
http://jipam.vu.edu.au/v1n1/

1.

Power-monotone sequences and Fourier series with positive coefficients

L. Leindler

2.

On Hadamard's Inequality on a Disk

S.S. Dragomir

3.

A Steffensen Type Inequality

Hillel Gauchman

4.

Generalized Abstracted Mean Values

Feng Qi

5.

An Inequality for Linear Positive Functionals

Bogdan Gavrea and Ioan Gavrea

6.

Inequalities for Planar Convex Sets

Paul R. Scott and Poh Wah Awyong

7.

Reverse Weighted Lp - Norm Inequalities in Convolutions

Saburou Saitoh, Vu Kim Tuan and Masahiro Yamamoto

8.

Existence and Local Uniqueness for Nonlinear Lidstone Boundary Value Problems

Jeffrey Ehme and Johnny Henderson

9.

On Hadamard's Inequality for the Convex Mappings Defined on a Convex Domain in the Space

Bogdan Gavrea

10.

Weighted Modular Inequalities for Hardy-Type Operators on Monotone Functions

Hans P. Heinig and Qinsheng Lai

 

Editors

R.P. Agarwal
G. Anastassiou
T. Ando
H. Araki
A.G. Babenko
D. Bainov
N.S. Barnett
H. Bor
J. Borwein
P.S. Bullen
P. Cerone
S.H. Cheng
L. Debnath
S.S. Dragomir
N. Elezovic
A.M. Fink
A. Fiorenza
T. Furuta
L. Gajek
H. Gauchman
C. Giordano
F. Hansen
D. Hinton
A. Laforgia
L. Leindler
C.-K. Li
L. Losonczi 
A. Lupas
R. Mathias
T. Mills
G.V. Milovanovic
R.N. Mohapatra
B. Mond
M.Z. Nashed
C.P. Niculescu
I. Olkin
B. Opic
B. Pachpatte
Z. Pales
C.E.M. Pearce
J. Pecaric
L.-E. Persson
L. Pick
I. Pressman
S. Puntanen
F. Qi
A.G. Ramm
T.M. Rassias
A. Rubinov
S. Saitoh
J. Sandor
S.P. Singh
A. Sofo
H.M. Srivastava
K.B. Stolarsky
G.P.H. Styan
L. Toth
R. Verma
F. Zhang

© 2000 School of Communications and Informatics, Victoria University of Technology. All rights reserved.
JIPAM is published by the School of Communications and Informatics which is part of the Faculty of Engineering and Science, located in Melbourne, Australia. All correspondence should be directed to the editorial office.

Copyright/Disclaimer


Up ] Article 1 ] Article 2 ] Article 3 ] Article 4 ] Article 5 ] Article 6 ] Article 7 ] Article 8 ] Article 9 ] [ Article 10 ]