This talk concerns $L^p$ bounds on the strong spherical maximal functions that are multi-parametric maximal functions defined by averages over ellipsoids. We obtain $L^p$ bounds on those maximal operators for a certain range of nontrivial p. No such maximal bounds have been known until recently. In particular, our results extend Stein’s celebrated spherical maximal bounds to multi-parametric versions. The talk is based on joint work with Juyoung Lee and Sewook Oh.

We consider a torus embedded in the 3-dimensional Euclidean space. It has a natural two-parameter scaling structure. Under this structure, we can consider a two parameter maximal average over the tori. We study the sharp boundedness of this maximal function, its Sobolev regularity, and local smoothing properties. We compare this result with the one parameter maximal function.

We consider two types of multilinear averaging operators over hypersurfaces with curvature conditions. We discuss $L^p$-improving estimates for these operators and $L^p$ estimates for the corresponding lacunary maximal operators. This talk is based on the joint work with Jin Bong Lee and Kalachand Shuin.

We define the atomic Hardy space $H^p_{\mathcal{L},\operatorname{at}}(\mathbb{C}^n)$, $0\lt p≤ 1$, for the twisted Laplacian $\mathcal{L}$ and prove its equivalence with the Hardy space defined using the maximal function corresponding to the heat semigroup $e^{-\mathcal{L}t}$, $t\gt 0$. We also prove sharp $L^p$, $0\lt p≤ 1$, estimates for $\mathcal{L}^{β/2}e^{i\sqrt{\mathcal{L}}}$. More precisely, we prove that it is a bounded operator on $H^p_{\mathcal{L},\operatorname{at}}(\mathbb{C}^n)$ when $β≥(2n-1)(1/p-1/2)$.

For measuring possible concentrations of the eigenfunctions of the Laplace operator on a manifold, Burq-Gerard-Tzvetkov studied $L^p$ norm of the restrictions of the eigenfunctions to submanifolds. They proved sharp $L^p$ estimates restricted to the geodesic or a curve having nonvanishing geodesic curvature. I will talk about $L^p$ estimates restricted to a curve which is not geodesic and has vanishing geodesic curvature. The proof involves semiclassical analysis.