Motivated by a recent work of Schippa [27], we consider local smoothing estimates for Schrödinger equations in modulation spaces. By using the Córdoba-Fefferman type reverse square function inequality and the bilinear Strichartz estimate, we can refine the summability exponent of modulation spaces. Next, we also discuss a new type of randomized Strichartz estimate in modulation spaces. Finally, we show that the reverse square function estimate implies the Strichartz estimates in modulation spaces. From this implication, we obtain the reverse square function estimate of critical order.
