Caltech/UCLA/USC Joint Analysis Seminar
UCLA, room MS 6221
We shall discuss sharp (up to end points) $L^p\to L^q$ estimates for local maximal operators associated with dilates of two different surfaces on Heisenberg groups. The first is the ``horizontal sphere" of codimension two. The second is the Kor\'anyi sphere: a surface of codimension one compatible with the non-isotropic dilation structure on the group but with points of vanishing curvature. We shall examine the geometry of these surfaces in light of two different notions of curvature and compare their effect on the estimates for the corresponding maximal operators. The Heisenberg group structure will play a crucial role in our arguments. However, the theory of Oscillatory Integral Operators will be central despite the non-Euclidean setting. We shall also discuss two new counterexamples which imply the sharpness of our results (up to endpoints). Partly based on joint work with Joris Roos and Andreas Seeger.