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Weighted Modular Inequalities for Hardy-Type Operators on Monotone Functions  
 
  Authors: Hans P. Heinig, Qin Sheng Lai,  
  Keywords: Hardy-type operators, modular inequalities, weights, N-functions, characterizations, Orlicz-Lorentz  
  Date Received: 03/11/99  
  Date Accepted: 31/01/00  
  Subject Codes:

26D15,42B25,26A33,46E30

  
  Editors: Bohumir Opic,  
 
  Abstract:

If

egin{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

egin{displaymath} Q^{-1}left( int_{0}^{infty }Q[ heta (Kf)]wight) leq P^{-1}left( int_{0}^{infty }P[Cf]vight) 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 $Qcirc P^{-1}$ is concave. Our results generalize those in [7,24], where the case $Qcirc P^{-1}$ 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.

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