The minimal operator and the geometric maximal operator in ${\Bbb R}^n$

David Cruz-Uribe, SFO¹

¹ Department of Mathematics Trinity College Hartford, CT 06106-3100, U.S.A.

Studia Mathematica, Tome 144 (2001) no. 1, pp. 1-37

Résumé

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 $M_0^*f=\mathop {\rm lim}_{r\rightarrow 0}M(|f|^r)^{1/r}$.

Détail

DOI : 10.4064/sm144-1-1

Keywords: prove two weight norm inequalities mathbb minimal operator large mathop inf int extending higher dimensions results obtained cruz uribe neugebauer olesen real line application extend mathbb weighted norm inequalities geometric maximal operator mathop sup mathop exp nolimits int mathop log nolimits right proved yin muckenhoupt norm inequalities centered minimal operator study powers doubling weights sufficient conditions geometric maximal operator equal closely related limiting operator *f mathop lim rightarrow