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Sharp Logarithmic Inequalities for Two Hardy-type Operators

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

Vol. 63, no. 3

237--247

For any locally integrable f on Rn, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: [WZÓR] for x ∈ Rn. The purpose of the paper is to establish sharp localized LlogL estimates for S and T. The proof rests on a corresponding one-weight estimate for a martingale maximal function, a result which is of independent interest.