On applicability of the sparse grid method in the worst case setting

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_k:W^{r_k}_p\to W^{s_k}_q$\end{document} be bounded linear operators. We provide several sufficient conditions for the validity of the inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|\otimes_kT_k\|\le\prod_k\|T_k\|$\end{document}. These results can be applied to error and cost estimates for the sparse grid method.