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• # Artykuł - szczegóły

## Studia Mathematica

2015 | 227 | 1 | 1-19

## Optimal estimates for the fractional Hardy operator

EN

### Abstrakty

EN
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 $A'_{α}$ 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 $A_{α}$.

1-19

wydano
2015

### Twórcy

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