We investigate the boundedness of the fractional maximal operator with respect to a general basis on weighted Lebesgue spaces. We characterize the boundedness of these operators for one-weight and two-weight inequalities extending the work of Jawerth. A new two-weight testing condition for the fractional maximal operator on a general basis is introduced extending the work of Sawyer for the basis of cubes. We also find the sharp dependence in the two-weight case between the operator norm and the testing condition of Sawyer. Finally, our approach leads to a new proof of Buckley's sharp estimate for the Hardy-Littlewood maximal function.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm194-2-4,
author = {Kabe Moen},
title = {Sharp one-weight and two-weight bounds for maximal operators},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {163-180},
zbl = {1174.42020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm194-2-4}
}
Kabe Moen. Sharp one-weight and two-weight bounds for maximal operators. Studia Mathematica, Tome 192 (2009) pp. 163-180. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm194-2-4/