Young’s Inequality for the Twisted Convolution

We characterise the triples (p, q, r) for which the bi-linear λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-twisted convolution map Bλ:(f,g)→f×λg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\lambda : (f,g) \rightarrow f \times _\lambda \, g$$\end{document} is bounded from Lp(Cn)×Lq(Cn)→Lr(Cn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p(\mathbb C^n) \times L^q(\mathbb C^n) \rightarrow L^r(\mathbb C^n)$$\end{document} for 1≤p,q,r≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le p,q,r \le \infty $$\end{document}. This gives the analogue of Young’s inequality for the twisted convolution.