Multidimensional Exponential Inequalities with Weights

Luor, Dah-Chin

Canadian mathematical bulletin, Tome 53 (2010) no. 2, pp. 327-339

Résumé

We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality $${{\left( \int_{E}{\Phi {{\left( {{T}_{k}}f\left( x \right) \right)}^{q}}u\left( x \right)dx} \right)}^{1/q}}\,\le \,C{{\left( \int_{E}{\Phi {{\left( f\left( x \right) \right)}^{p}}v\left( x \right)dx} \right)}^{1/p}},$$ where $0\,<\,p,\,q\,<\,\infty $ , $\Phi $ is a logarithmically convex function, and ${{T}_{k}}$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin–Cochran-Lee type inequalities.

Détail

DOI : 10.4153/CMB-2010-038-1

Mots-clés : 26D15, 26D10, multidimensional inequalities, geometric mean operators, exponential inequalities, star-shaped regions