In the field of Euclidean harmonic analysis, two of the main research topics are the convergence of Fourier series, and $L^p$ boundedness of maximal averages which corresponds to the Lebesgue differentiation theory. One classical maximal averaging operator is the spherical maximal operator (also known as Stein’s maximal operator). In this presentation, we will explore the convergence of product of Fourier series. Specifically, our focus will be on examining the boundedness in $L^p$ and weighted $L^p$ spaces of bilinear Bochner–Riesz means.
Additionally, we will delve into the $L^p$ boundedness of linear and bilinear maximal averages defined on hypersurfaces where the Gaussian curvature vanishes. Our findings reveal that having a nonvanishing Gaussian curvature alone does not suffice, in contrast to the $L^p$-boundedness of the linear maximal operator associated with hypersurfaces, to fully characterize the optimal maximal boundedness.
