Sharp Logarithmic Inequalities for Two Hardy-type Operators

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.