Abstract :

Maximal functions are of great interest in Harmonic Analysis, PDEs, Ergodic theory, and other areas of Mathematics, dating back to the time of Hardy and Littlewood. In this talk, we shall explore three variants of maximal functions, focusing primarily on the maximal wave propagator and maximal functions on surfaces. A significant portion of this talk will be dedicated to their study on the Hardy spaces for Fourier integral operators. Our estimates on the Hardy spaces for Fourier integral operators will imply new and improved pointwise convergence results for the wave propagator. Finally, we will briefly discuss a third variant: a maximal function on an “exponentially” growing metric measure space. Based on a joint work with J. Rozendaal, N. Liu, and L. Song.