Weighted Inequalities for Hardy–Steklov Operators

Bernardis, A. L. ; Martín-Reyes, F. J. ; Salvador, P. Ortega

Canadian journal of mathematics, Tome 59 (2007) no. 2, pp. 276-295

Résumé

We characterize the pairs of weights $\left( v,\,w \right)$ for which the operator $Tf\left( x \right)=g\left( x \right)\int{_{s\left( x \right)}^{h\left( x \right)}}\,f\,$ with $s$ and $h$ increasing and continuous functions is of strong type $\left( p,\,q \right)$ or weak type $\left( p,\,q \right)$ with respect to the pair $\left( v,\,w \right)$ in the case $0\,<\,q\,<\,p$ and $1\,<\,p\,<\,\infty$ . The result for the weak type is new while the characterizations for the strong type improve the ones given by H. P. Heinig and G. Sinnamon. In particular, we do not assume differentiability properties on $s$ and $h$ and we obtain that the strong type inequality $\left( p,q \right),q\,<p$ , is characterized by the fact that the function $$\Phi \left( x \right)\,=\,\sup \,{{\left( \int_{c}^{d}{{{g}^{q}}w} \right)}^{1/p}}\left( \int_{s\left( d \right)}^{h\left( c \right)}{{{v}^{1-{p}'}}} \right){{\,}^{1/{p}'}}$$ belongs to ${{L}^{r}}\left( {{g}^{q}}w \right)$ , where ${1}/{r\,=\,{\,1}/{q\,-\,{1}/{p}\;}\;}\;$ and the supremum is taken over all $c$ and $d$ such that $c\le x\le d$ and $s\left( d \right)\,\le \,h\left( c \right)$ .

Détail

DOI : 10.4153/CJM-2007-011-x

Mots-clés : 26D15, 46E30, 42B25, Hardy–Steklov operator, weights, inequalities

Bibliographie

[1] [1] Bernardis, A. L., Martín-Reyes, F. J. and Ortega Salvador, P., A new proof of the characterization of the weighted Hardy inequality. Proc. Roy. Soc. Edinburgh Sect. A 135(2005), no. 5, 941–945. Google Scholar

[2] [2] Chen, T. and Sinnamon, G., Generalized Hardy operators and normalizing measures. J. Inequal. Appl. 7(2002), no. 6, 829–866. Google Scholar

[3] [3] Gogatishvili, A. and Lang, J., The generalized Hardy operator with kernel and variable integral limits in Banach function spaces. J. Inequal. Appl. 4(1999), no. 1, 1–16. Google Scholar

[4] [4] Heinig, H. P. and Sinnamon, G., Mapping properties of integral averaging operators. Studia Math. 129(1998), no. 2, 157–177. Google Scholar

[5] [5] Kufner, A. and Persson, L. E., Weighted Inequalities of Hardy type. World Scientific, Riveredge, NJ, 2003. Google Scholar

[6] [6] Lai, Q., Weighted modular inequalities for Hardy type operators. Proc. London Math. Soc. 79(1999), 649–672 Google Scholar

[7] [7] Martín-Reyes, F. J. and Ortega, P., On weighted weak type inequalities for modified Hardy operators. Proc. Amer. Math. Soc. 126(1998), no. 6, 1739–1746. Google Scholar

[8] [8] Maz’ja, W. G., Sobolev Spaces. Springer-Verlag, Berlin, 1985. Google Scholar

[9] [9] Sawyer, E., Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. Trans. Amer. Math. Soc. 281(1984), no. 1, 329–337. Google Scholar