Certain weighted norm inequalities for integral operators with non-negative, monotone kernels are shown to remain valid when the weight is replaced by a monotone majorant or minorant of the original weight. A similar result holds for operators with quasi-concave kernels. To prove these results a careful investigation of the functional properties of the monotone envelopes of a non-negative function is carried-out. Applications are made to function space embeddings of the cones of monotone functions and quasi-concave functions. Under weaker partial orders on non-negative functions, monotone envelopes are re-examined and the level function is recognized as a monotone envelope in two ways. Using the level function, monotonicity can be transferred from the kernel to the weight in inequalities restricted to a cone of monotone functions.
@article{urn:eudml:doc:43083,
title = {Transferring monotonicity in weighted norm inequalities.},
journal = {Collectanea Mathematica},
volume = {54},
year = {2003},
pages = {181-216},
zbl = {1093.26025},
mrnumber = {MR1995140},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:43083}
}
Sinnamon, Gord. Transferring monotonicity in weighted norm inequalities.. Collectanea Mathematica, Tome 54 (2003) pp. 181-216. http://gdmltest.u-ga.fr/item/urn:eudml:doc:43083/