A Generalization of the Hausdorff-Young Theorem

Considering mixed-norm sequence spaces lp,q, p, q ≧ 1, C. N. Kellogg proved the following theorem: if 1 < p ≦ 2 then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\widehat{L^p }$$ \end{document} ⊂ lp′,2 and lp,2 ⊂ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\widehat{L^{p'} }$$ \end{document}, where 1/p + 1/p′ = 1. This result extends the Hausdorff-Young Theorem.