Sharp Logarithmic Inequalities for Two Hardy-type Operators

Adam Osękowski¹

¹ Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 63 (2015) no. 3, pp. 237-247

Résumé

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.

Détail

DOI : 10.4064/ba8039-12-2015

Keywords: locally integrable mathbb consider operators which average balls radius center respectively frac int hskip frac int mathbb purpose paper establish sharp localized llogl estimates proof rests corresponding one weight estimate martingale maximal function result which independent interest