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

Tom 144 / 2001

David Cruz-Uribe, SFO Studia Mathematica 144 (2001), 1-37 MSC: Primary 42B25. DOI: 10.4064/sm144-1-1

Streszczenie

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

Autorzy

  • David Cruz-Uribe, SFODepartment of Mathematics
    Trinity College
    Hartford, CT 06106-3100, U.S.A.
    e-mail

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