The minimal operator and the geometric maximal operator in ${\Bbb R}^n$
Volume 144 / 2001
Abstract
We prove two-weight norm inequalities in ${\mathbb R}^n$ for the minimal operator $$ {\Large m}f(x) = \mathop {\rm inf}_{Q\ni x} {1\over |Q|} \int _Q |f|\, dy, $$ extending to higher dimensions results obtained by Cruz-Uribe, Neugebauer and Olesen [8] on the real line. As an application we extend to ${\mathbb R}^n$ weighted norm inequalities for the geometric maximal operator $$ M_0f(x) = \mathop {\rm sup}_{Q\ni x}\mathop {\rm exp}\nolimits \left ({1\over |Q|}\int _Q \mathop {\rm log}\nolimits |f|\, dx \right ), $$ proved by Yin and Muckenhoupt [27].
We also give norm inequalities for the centered minimal operator, study powers of doubling weights and give sufficient conditions for the geometric maximal operator to be equal to the closely related limiting operator $M_0^*f=\mathop {\rm lim}_{r\rightarrow 0}M(|f|^r)^{1/r}$.
