Optimal estimates for the fractional Hardy operator

Yoshihiro Mizuta, Aleš Nekvinda, and Tetsu Shimomura. "Optimal estimates for the fractional Hardy operator." Studia Mathematica 227.1 (2015): 1-19. <http://eudml.org/doc/286248>.

@article{YoshihiroMizuta2015,
abstract = {Let $A_\{α\}f(x) = |B(0,|x|)|^\{-α/n\} ∫_\{B(0,|x|)\} f(t)dt$ be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that $A_\{α\}$ is bounded from $L^\{p\}$ to $L^\{p_\{α\}\}$ with $p_\{α\} = np/(αp-np+n)$ when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space $S_\{α,Y\}$, which is strictly larger than X, and a ’target’ space $T_\{Y\}$, which is strictly smaller than Y, under the assumption that $A_\{α\}$ is bounded from X into Y and the Hardy-Littlewood maximal operator M is bounded from Y into Y, and prove that $A_\{α\}$ is bounded from $S_\{α,Y\}$ into $T_\{Y\}$. We prove optimality results for the action of $A_\{α\}$ and the associate operator $A^\{\prime \}_\{α\}$ on such spaces, as an extension of the results of Mizuta et al. (2013) and Nekvinda and Pick (2011). We also study the duals of optimal spaces for $A_\{α\}$.},
author = {Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura},
journal = {Studia Mathematica},
keywords = {fractional Hardy operator; Banach function space; optimal spaces; weighted Lebesgue spaces; Lorentz spaces},
language = {eng},
number = {1},
pages = {1-19},
title = {Optimal estimates for the fractional Hardy operator},
url = {http://eudml.org/doc/286248},
volume = {227},
year = {2015},
}

TY - JOUR
AU - Yoshihiro Mizuta
AU - Aleš Nekvinda
AU - Tetsu Shimomura
TI - Optimal estimates for the fractional Hardy operator
JO - Studia Mathematica
PY - 2015
VL - 227
IS - 1
SP - 1
EP - 19
AB - Let $A_{α}f(x) = |B(0,|x|)|^{-α/n} ∫_{B(0,|x|)} f(t)dt$ be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that $A_{α}$ is bounded from $L^{p}$ to $L^{p_{α}}$ with $p_{α} = np/(αp-np+n)$ when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space $S_{α,Y}$, which is strictly larger than X, and a ’target’ space $T_{Y}$, which is strictly smaller than Y, under the assumption that $A_{α}$ is bounded from X into Y and the Hardy-Littlewood maximal operator M is bounded from Y into Y, and prove that $A_{α}$ is bounded from $S_{α,Y}$ into $T_{Y}$. We prove optimality results for the action of $A_{α}$ and the associate operator $A^{\prime }_{α}$ on such spaces, as an extension of the results of Mizuta et al. (2013) and Nekvinda and Pick (2011). We also study the duals of optimal spaces for $A_{α}$.
LA - eng
KW - fractional Hardy operator; Banach function space; optimal spaces; weighted Lebesgue spaces; Lorentz spaces
UR - http://eudml.org/doc/286248
ER -