2018/10/14 by Jungjin Lee, Lee, Jungjin
Mathematics · #35B65 #42B15 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1810.06041
openalex publication_date 2018/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we show that the local Kato type smoothing estimates are\nessentially equivalent to the global Kato type smoothing estimates for some\nclass of dispersive equations including the Schr "odinger equation. From this\nwe immediately have two results as follows. One is that the known local Kato\nsmoothing estimates are sharp. The sharp regularity ranges of the global Kato\nsmoothing estimates are already known, but those of the local Kato smoothing\nestimates are not. Recently, Sun, Tr 'elat, Zhang and Zhong have shown it only\nin spacetime mathbb R \× mathbb R. Our result resolves this issue in\nhigher dimensions. The other one is the sharp global-in-time maximal\nSchr "odinger estimates. Recently, the pointwise convergence conjecture of the\nSchr "odinger equation has been settled by Du--Guth--Li--Zhang and Du--Zhang.\nFor this they proved related sharp local maximal Schr "odinger estimates. By\nour result, these lead to the sharp global-in-time maximal Schr "odinger\nestimates.\n