Sharp one-weight and two-weight bounds for maximal operators

Kabe Moen ¹

¹ Department of Mathematics University of Kansas 405 Snow Hall 1460 Jayhawk Blvd. Lawrence, KS 66045-7523, U.S.A.

Studia Mathematica, Tome 194 (2009) no. 2, pp. 163-180

Résumé

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.

Détail

DOI : 10.4064/sm194-2-4

Keywords: investigate boundedness fractional maximal operator respect general basis weighted lebesgue spaces characterize boundedness these operators one weight two weight inequalities extending work jawerth two weight testing condition fractional maximal operator general basis introduced extending work sawyer basis cubes sharp dependence two weight between operator norm testing condition sawyer finally approach leads proof buckleys sharp estimate hardy littlewood maximal function