{"title":"带多项式符号的周期性分散方程收敛问题的反例","authors":"Daniel Eceizabarrena, Xueying Yu","doi":"arxiv-2408.13935","DOIUrl":null,"url":null,"abstract":"In the setting of Carleson's convergence problem for the fractional\nSchr\\\"odinger equation $i\\, \\partial_t u + (-\\Delta)^{a/2}u=0$ with $a > 1$ in\n$\\mathbb R^d$, which has Fourier symbol $P(\\xi) = |\\xi|^a$, it is known that\nthe Sobolev exponent $d/(2(d+1))$ is sufficient, but it is not known whether\nthis condition is necessary. In this article, we show that in the periodic\nproblem in $\\mathbb T^d$ the exponent $d/(2(d+1))$ is necessary for all\nnon-singular polynomial symbols $P$ regardless of the degree of $P$. Among the\ndifferential operators covered, we highlight the natural powers of the\nLaplacian $\\Delta^k$ for $k \\in \\mathbb N$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"3 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Counterexamples to the convergence problem for periodic dispersive equations with a polynomial symbol\",\"authors\":\"Daniel Eceizabarrena, Xueying Yu\",\"doi\":\"arxiv-2408.13935\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In the setting of Carleson's convergence problem for the fractional\\nSchr\\\\\\\"odinger equation $i\\\\, \\\\partial_t u + (-\\\\Delta)^{a/2}u=0$ with $a > 1$ in\\n$\\\\mathbb R^d$, which has Fourier symbol $P(\\\\xi) = |\\\\xi|^a$, it is known that\\nthe Sobolev exponent $d/(2(d+1))$ is sufficient, but it is not known whether\\nthis condition is necessary. In this article, we show that in the periodic\\nproblem in $\\\\mathbb T^d$ the exponent $d/(2(d+1))$ is necessary for all\\nnon-singular polynomial symbols $P$ regardless of the degree of $P$. Among the\\ndifferential operators covered, we highlight the natural powers of the\\nLaplacian $\\\\Delta^k$ for $k \\\\in \\\\mathbb N$.\",\"PeriodicalId\":501145,\"journal\":{\"name\":\"arXiv - MATH - Classical Analysis and ODEs\",\"volume\":\"3 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Classical Analysis and ODEs\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.13935\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Classical Analysis and ODEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.13935","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Counterexamples to the convergence problem for periodic dispersive equations with a polynomial symbol
In the setting of Carleson's convergence problem for the fractional
Schr\"odinger equation $i\, \partial_t u + (-\Delta)^{a/2}u=0$ with $a > 1$ in
$\mathbb R^d$, which has Fourier symbol $P(\xi) = |\xi|^a$, it is known that
the Sobolev exponent $d/(2(d+1))$ is sufficient, but it is not known whether
this condition is necessary. In this article, we show that in the periodic
problem in $\mathbb T^d$ the exponent $d/(2(d+1))$ is necessary for all
non-singular polynomial symbols $P$ regardless of the degree of $P$. Among the
differential operators covered, we highlight the natural powers of the
Laplacian $\Delta^k$ for $k \in \mathbb N$.