The minimal operator and the geometric maximal operator in ℝⁿ

David Cruz-Uribe, SFO. "The minimal operator and the geometric maximal operator in ℝⁿ." Studia Mathematica 144.1 (2001): 1-37. <http://eudml.org/doc/284790>.

@article{DavidCruz2001,
abstract = {We prove two-weight norm inequalities in ℝⁿ for the minimal operator $ f(x) = inf_\{Q∋x\} 1/|Q| ∫_\{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 ℝⁿ weighted norm inequalities for the geometric maximal operator $M₀f(x) = \{sup\}_\{Q∋x\} exp(1/|Q| ∫_\{Q\} log|f|dx)$, 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 $M₀*f = lim_\{r→0\} M(|f|^\{r\})^\{1/r\}$.},
author = {David Cruz-Uribe, SFO},
journal = {Studia Mathematica},
keywords = {minimal operator; geometric maximal operator; weighted norm inequalities},
language = {eng},
number = {1},
pages = {1-37},
title = {The minimal operator and the geometric maximal operator in ℝⁿ},
url = {http://eudml.org/doc/284790},
volume = {144},
year = {2001},
}

TY - JOUR
AU - David Cruz-Uribe, SFO
TI - The minimal operator and the geometric maximal operator in ℝⁿ
JO - Studia Mathematica
PY - 2001
VL - 144
IS - 1
SP - 1
EP - 37
AB - We prove two-weight norm inequalities in ℝⁿ for the minimal operator $ f(x) = inf_{Q∋x} 1/|Q| ∫_{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 ℝⁿ weighted norm inequalities for the geometric maximal operator $M₀f(x) = {sup}_{Q∋x} exp(1/|Q| ∫_{Q} log|f|dx)$, 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 $M₀*f = lim_{r→0} M(|f|^{r})^{1/r}$.
LA - eng
KW - minimal operator; geometric maximal operator; weighted norm inequalities
UR - http://eudml.org/doc/284790
ER -