Optimal estimates for the fractional Hardy operator

Yoshihiro Mizuta ; Aleš Nekvinda ; Tetsu Shimomura

Studia Mathematica, Tome 231 (2015), p. 1-19 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $A_{α}$.

Détail

Publié le : 2015-01-01

Zbl 1328.47049

EUDML-ID : urn:eudml:doc:286248