Sharp bounds for the Schr\"odinger maximal function on the torus

发布时间：2025-11-06 作者：浏览次数：

Speaker:	张再云	DateTime:	2025年11月7日（周五）下午15：30-17：00
Brief Introduction to Speaker: 张再云教授，湖南理工学院		Place:	国交2号楼201会议室
Abstract:In this talk, we investigate the sharp bounds for the Schr\"odinger maximal function on the torus. First, using harmonic analysis method and cut-off function technique, we establish the bilinear restriction estimate on the tours. Second, using the Littlewood-Paley decomposition, we obtain the maximal estimate for the quartic Weyl sums. Then, using the bilinear restriction estimate and the maximal estimate for quartic Weyl sums, we demonstrate the bound on $\sup\limits_{t}\|u(x,t)\|$ for the solutions $u$ to the four order Schr\"odinger equation on the torus. Finally, using the method of exponential sums and the Hardy-Littlewood circle method, basic number theory, Diophantine analysis \cite{DZ19} as well as the major arc estimate, we show that the maximal estimate is sharp for the quartic Weyl sums, up to the loss $N^{\varepsilon}.$