Radial multipliers and restriction to surfaces of the Fourier transform in mixed-norm spaces

In this article we revisit some classical conjectures in harmonic analysis in the setting of mixed norm spaces LradpLang2Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{rad}^{p}L_{ang}^{2}\left( \mathbb {R}^{n}\right) $$\end{document}. We produce sharp bounds for the restriction of the Fourier transform to compact hypersurfaces of revolution in the mixed norm setting and study an extension of the disc multiplier. We also present some results for the discrete restriction conjecture and state an intriguing open problem.