Optimal estimates for the fractional Hardy operator

Yoshihiro Mizuta¹ ; Aleš Nekvinda² ; Tetsu Shimomura³

¹ Department of Mechanical Systems Engineering Hiroshima Institute of Technology 2-1-1 Miyake Saeki-ku, Hiroshima 731-5193, Japan
² Faculty of Civil Engineering Czech Technical University Thákurova 7 166 29 Praha 6, Czech Republic
³ Department of Mathematics Graduate School of Education Hiroshima University Higashi-Hiroshima 739-8524, Japan

Studia Mathematica, Tome 227 (2015) no. 1, pp. 1-19

Résumé

Let $A_{\alpha}f(x) = |B(0,|x|)|^{-\alpha/n} \int_{B(0,|x|)} f(t) \,dt$ be the $n$-dimensional fractional Hardy operator, where $0\alpha \le n$. It is well-known that $A_{\alpha}$ is bounded from $L^p$ to $L^{p_\alpha}$ with $p_\alpha=np/(\alpha p-np+n)$ when $n(1-1/p)\alpha \le n$. 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 $S_{\alpha,Y}$, which is strictly larger than $X$, and a `target' space $T_Y$, which is strictly smaller than $Y$, under the assumption that $A_{\alpha}$ is bounded from $X$ into $Y$ and the Hardy–Littlewood maximal operator $M$ is bounded from $Y$ into $Y$, and prove that $A_{\alpha}$ is bounded from $S_{\alpha,Y}$ into $T_Y$. We prove optimality results for the action of $A_{\alpha}$ and the associate operator $A'_\alpha$ 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 $A_\alpha$.

Détail

DOI : 10.4064/sm227-1-1

Keywords: alpha alpha int n dimensional fractional hardy operator where alpha well known alpha bounded alpha alpha alpha p np alpha improve result within framework banach function spaces instance weighted lebesgue spaces lorentz spaces source space alpha which strictly larger nbsp target space nbsp which strictly smaller nbsp under assumption alpha bounded hardy littlewood maximal operator bounded prove alpha bounded alpha prove optimality results action alpha associate operator alpha spaces extension results mizuta nekvinda pick study duals optimal spaces nbsp alpha