Bounds for Lacunary Bilinear Spherical and Triangle Maximal Functions

We prove Lp(Rd)xLq(Rd)-> Lr(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>p({\mathbb {R}}<^>d)\times L<^>q({\mathbb {R}}<^>d)\rightarrow L<^>r({\mathbb {R}}<^>d)$$\end{document} bounds for certain lacunary bilinear maximal averaging operators with parameters satisfying the H & ouml;lder relation 1/p+1/q=1/r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/p+1/q=1/r$$\end{document}. The boundedness region that we get contains at least the interior of the H & ouml;lder boundedness region of the associated single scale bilinear averaging operator. In the case of the lacunary bilinear spherical maximal function in d >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 2$$\end{document}, we prove boundedness for any p,q is an element of(1,infinity]2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p,q\in (1,\infty ]<^>2$$\end{document}, which is sharp up to boundary; we then show how to extend this result to a more degenerate family of surfaces where some curvatures are allowed to vanish. For the lacunary triangle averaging maximal operator, we have results in d >= 7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 7$$\end{document}, and the description of the sharp region will depend on a sharp description of the H & ouml;lder bounds for the single scale triangle averaging operator, which is still open.