Sharp Logarithmic Inequalities for Two Hardy-type Operators

Adam Osękowski. "Sharp Logarithmic Inequalities for Two Hardy-type Operators." Bulletin of the Polish Academy of Sciences. Mathematics 63.3 (2015): 237-247. <http://eudml.org/doc/281161>.

@article{AdamOsękowski2015,
abstract = {For any locally integrable f on ℝⁿ, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: $Sf(x) = 1/|B(0,|x|)| ∫_\{B(0,|x|)\} f(t)dt$, $Tf(x) = 1/|B(x,|x|)| ∫_\{B(x,|x|)\} f(t)dt$ for x ∈ ℝⁿ. 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.},
author = {Adam Osękowski},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {Hardy-Littlewood operators; logarithmic inequalities; martingale maximal function},
language = {eng},
number = {3},
pages = {237-247},
title = {Sharp Logarithmic Inequalities for Two Hardy-type Operators},
url = {http://eudml.org/doc/281161},
volume = {63},
year = {2015},
}

TY - JOUR
AU - Adam Osękowski
TI - Sharp Logarithmic Inequalities for Two Hardy-type Operators
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2015
VL - 63
IS - 3
SP - 237
EP - 247
AB - For any locally integrable f on ℝⁿ, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: $Sf(x) = 1/|B(0,|x|)| ∫_{B(0,|x|)} f(t)dt$, $Tf(x) = 1/|B(x,|x|)| ∫_{B(x,|x|)} f(t)dt$ for x ∈ ℝⁿ. 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.
LA - eng
KW - Hardy-Littlewood operators; logarithmic inequalities; martingale maximal function
UR - http://eudml.org/doc/281161
ER -