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Zeitschrift für Analysis und ihre Anwendungen


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Volume 15, Issue 2, 1996, pp. 309–328
DOI: 10.4171/ZAA/702

Published online: 1996-06-30

Weighted Inequalities for the Fractional Maximal Operator and the Fractional Integral Operator

Y. Rakotondratsimba[1]

(1) Institut Polytechnique St. Louis, Cergy-Pontoise, France

A sufficient condition is given on weight functions $u$ and $v$ on $\mathbb R^n$ for which the fractional maximal operator $M_s (0 ≤ s < n)$ defined by $(M_sf)(x) = \mathrm {sup}_{Q \ni x} |Q|^{\frac{s}{n}–1} \int_Q | f (y) | dy$ or the fractional integral operator $I_s (0 < s < n)$ defined by $(I_s,f)(x) = \int_{\mathbb R^n} | x - y |^{s–n} f(y)dy$ is bounded from $L^p (\mathbb R^n, vdx)$ into $L^q(\mathbb R^n,udx)$ for $0 < q < p$ with $p> 1$, where $Q$ is a cube and n a non-negative integer.

Keywords: Weigthed inequalities, fractional maximal operators, fractional integral operators

Rakotondratsimba Y.: Weighted Inequalities for the Fractional Maximal Operator and the Fractional Integral Operator. Z. Anal. Anwend. 15 (1996), 309-328. doi: 10.4171/ZAA/702