{"title":"On gradient estimates for heat kernels","authors":"Baptiste Devyver","doi":"10.1080/03605302.2020.1857398","DOIUrl":null,"url":null,"abstract":"Abstract We study pointwise and Lp gradient estimates of the heat kernels of both the scalar Laplacian, as well as the Hodge Laplacian on k-forms, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. Such heat kernel estimates have already been obtained by the author, together with Coulhon and Sikora, provided certain L 2-cohomology spaces are trivial. This is however a strong topological assumption, and it is desirable to weaken it. The main point of the current work is to investigate what happens when these L 2-cohomology spaces are non-trivial. We find that the answer depends on some Lq integrability properties of L 2-harmonic forms.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"46 1","pages":"717 - 779"},"PeriodicalIF":2.1000,"publicationDate":"2021-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/03605302.2020.1857398","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Partial Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/03605302.2020.1857398","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract We study pointwise and Lp gradient estimates of the heat kernels of both the scalar Laplacian, as well as the Hodge Laplacian on k-forms, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. Such heat kernel estimates have already been obtained by the author, together with Coulhon and Sikora, provided certain L 2-cohomology spaces are trivial. This is however a strong topological assumption, and it is desirable to weaken it. The main point of the current work is to investigate what happens when these L 2-cohomology spaces are non-trivial. We find that the answer depends on some Lq integrability properties of L 2-harmonic forms.
期刊介绍:
This journal aims to publish high quality papers concerning any theoretical aspect of partial differential equations, as well as its applications to other areas of mathematics. Suitability of any paper is at the discretion of the editors. We seek to present the most significant advances in this central field to a wide readership which includes researchers and graduate students in mathematics and the more mathematical aspects of physics and engineering.