ABSTRACT

Sogge generalized the definitions of Nikodym sets and Nikodym maximal functions to Riemannian manifolds. He obtained a lower bound for the Minkowski dimensions of the Nikodym sets under two completely different curvature conditions: constant sectional curvature condition and chaotic curvature condition. This dichotomy result is reminiscent of Bourgain's condition on phase functions in oscillatory integrals. So, a natural question is how to formulate Bourgain's condition on distance functions of manifolds? And in contrast, how to formulate Sogge's chaotic curvature condition on phase or distance functions? We will discuss about these two questions in the talk. This is a joint work with Song Dai, Shaoming Guo and Ruixiang Zhang.