Noncommutative Bourgain's circular maximal theorem and a local smoothing estimate on the generalized Moyal planes

In this paper, we establish a local smoothing estimate on two-dimensional quantum Euclidean space. This is the noncommutative analogue of the one due to Mockenhaupt$-$Seeger$-$Sogge \cite{MSS}. As an application and simultaneously one motivation, we obtain the noncommutative analogue of Bourgain’s circular maximal theorem, resolving one problem after \cite{Hong}.