Geodesic
On conditions for the boundedness of the Weyl fractional integral on weighted $L^p$ spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 1, pp. 17-36 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this paper we give a sufficient condition on the pair of weights $(w,v)$ for the boundedness of the Weyl fractional integral $I_{\alpha}^+$ from $L^p(v)$ into $L^p(w)$. Under some restrictions on $w$ and $v$, this condition is also necessary. Besides, it allows us to show that for any $p: 1 \leq p \infty $ there exist non-trivial weights $w$ such that $I_{\alpha}^+$ is bounded from $L^p(w)$ into itself, even in the case $\alpha > 1$.
In this paper we give a sufficient condition on the pair of weights $(w,v)$ for the boundedness of the Weyl fractional integral $I_{\alpha}^+$ from $L^p(v)$ into $L^p(w)$. Under some restrictions on $w$ and $v$, this condition is also necessary. Besides, it allows us to show that for any $p: 1 \leq p \infty $ there exist non-trivial weights $w$ such that $I_{\alpha}^+$ is bounded from $L^p(w)$ into itself, even in the case $\alpha > 1$.
Classification : 26A33, 42B25, 44A15
Keywords: Weyl fractional integrals; weights 
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     author = {de Rosa, L. and de la Torre, A.},
     title = {On conditions for the boundedness of the {Weyl} fractional integral on weighted $L^p$ spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {17--36},
     year = {2004},
     volume = {45},
     number = {1},
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     zbl = {1100.26003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2004_45_1_a2/}
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de Rosa, L.; de la Torre, A. On conditions for the boundedness of the Weyl fractional integral on weighted $L^p$ spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 1, pp. 17-36. http://geodesic.mathdoc.fr/item/CMUC_2004_45_1_a2/