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Zeitschrift für Analysis und ihre Anwendungen

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Volume 35, Issue 1, 2016, pp. 1–20
DOI: 10.4171/ZAA/1552

Published online: 2015-12-23

Sharp Logarithmic Inequalities for Hardy Operators

Adam Osękowski[1]

(1) University of Warsaw, Poland

Let $\ell\geq 1$ be a fixed number. We determine, for each $K>0$, the best constant $L=L(K,\ell)\in (0,\infty]$ such that the following holds. If $f$ is a function on $(0,1]$ with $\int_0^1 |f(r)|\mbox{d}r=1$, then $$ \int_0^1 t^{\ell-1}\left(\frac{1}{t}\int_0^t |f(r)|\mbox{d}r\right)^\ell\mbox{d}t\leq K \int_0^1 |f(r)|\log |f(r)|\,\mbox{d}r+L.$$ As an application, we derive a sharp local logarithmic estimate for $n$-dimensional fractional Hardy operator.

Keywords: Maximal operator, fractional maximal operator, LlogL inequality, best constant

Osękowski Adam: Sharp Logarithmic Inequalities for Hardy Operators. Z. Anal. Anwend. 35 (2016), 1-20. doi: 10.4171/ZAA/1552