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Sharp Logarithmic Inequalities for Two Hardy-type Operators

Volume 63 / 2015

Adam Osękowski Bulletin Polish Acad. Sci. Math. 63 (2015), 237-247 MSC: Primary 42B10, 60G44; Secondary 46E30. DOI: 10.4064/ba8039-12-2015 Published online: 3 December 2015

Abstract

For any locally integrable $f$ on $\mathbb {R}^n$, we consider the operators $S$ and $T$ which average $f$ over balls of radius $|x|$ and center $0$ and $x$, respectively: $$ Sf(x)=\frac {1}{|B(0,|x|)|}\int _{B(0,|x|)} f(t)\,dt,\hskip 1em Tf(x)=\frac {1}{|B(x,|x|)|}\int _{B(x,|x|)} f(t)\,dt $$ for $x\in \mathbb {R}^n$. 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.

Authors

  • Adam OsękowskiDepartment of Mathematics, Informatics and Mechanics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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