Estimates for maximal functions associated to hypersurfaces in ℝ³ with height h < 2: Part I

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dc.contributor.author Buschenhenke, Stefan
dc.contributor.author Dendrinos, Spyridon
dc.contributor.author Ikromov, Isroil A.
dc.contributor.author Müller, Detlef
dc.date.accessioned 2019-08-30T10:22:35Z
dc.date.available 2019-08-30T10:22:35Z
dc.date.issued 2019-07-15
dc.identifier.citation Buschenhenke, S., Dendrinos, S., Ikromov, I. A. and Müller, D. (2019) 'Estimates for maximal functions associated to hypersurfaces in ℝ³ with height h < 2: Part I', Transactions of the American Mathematical Society, 372(2), pp. 1363-1406. doi: 10.1090/tran/7633 en
dc.identifier.volume 372 en
dc.identifier.issued 2 en
dc.identifier.startpage 1363 en
dc.identifier.endpage 1406 en
dc.identifier.issn 0002-9947
dc.identifier.uri http://hdl.handle.net/10468/8418
dc.identifier.doi 10.1090/tran/7633 en
dc.description.abstract In this article, we continue the study of the problem of Lp-boundedness of the maximal operator M associated to averages along isotropic dilates of a given, smooth hypersurface S of finite type in 3-dimensional Euclidean space. An essentially complete answer to this problem was given about eight years ago by the third and fourth authors in joint work with M. Kempe [Acta Math 204 (2010), pp. 151–271] for the case where the height h of the given surface is at least two. In the present article, we turn to the case h < 2. More precisely, in this Part I, we study the case where h < 2, assuming that S is contained in a sufficiently small neighborhood of a given point x0 ∈ S at which both principal curvatures of S vanish. Under these assumptions and a natural transversality assumption, we show that, as in the case h ≥ 2, the critical Lebesgue exponent for the boundedness of M remains to be pc = h, even though the proof of this result turns out to require new methods, some of which are inspired by the more recent work by the third and fourth authors on Fourier restriction to S. Results on the case where h < 2 and exactly one principal curvature of S does not vanish at x0 will appear elsewhere. en
dc.description.sponsorship Deutsche Forschungsgemeinschaft (DFG-Grant MU 761/11-1); National Science Foundation (Grant No. DMS-1440140) en
dc.format.mimetype application/pdf en
dc.language.iso en en
dc.publisher American Mathematical Society en
dc.rights © 2019, American Mathematical Society. First published in Transactions of the American Mathematical Society 372(2), July 2019. This author's draft version of the work, after peer review, is reproduced under the CC BY-NC-ND Creative Commons License. en
dc.rights.uri https://creativecommons.org/licenses/by-nc-nd/4.0/
dc.subject Smooth hypersurface en
dc.subject 3-dimensional Euclidean space en
dc.subject Isotropic dilates en
dc.title Estimates for maximal functions associated to hypersurfaces in ℝ³ with height h < 2: Part I en
dc.type Article (peer-reviewed) en
dc.internal.authorcontactother Spyridon Dendrinos, School Of Mathematical Sciences, University College Cork, Cork, Ireland. +353-21-490-3000 Email: sd@ucc.ie en
dc.internal.availability Full text available en
dc.date.updated 2019-08-30T09:58:06Z
dc.description.version Accepted Version en
dc.internal.rssid 464800032
dc.contributor.funder Deutsche Forschungsgemeinschaft en
dc.contributor.funder National Science Foundation en
dc.contributor.funder Mathematical Sciences Research Institute, Berkeley, California en
dc.description.status Peer reviewed en
dc.identifier.journaltitle Transactions of the American Mathematical Society en
dc.internal.copyrightchecked Yes
dc.internal.licenseacceptance Yes en
dc.internal.IRISemailaddress sd@ucc.ie en
dc.identifier.eissn 1088-6850


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© 2019, American Mathematical Society. First published in Transactions of the American Mathematical Society 372(2), July 2019. This author's draft version of the work, after peer review, is reproduced under the CC BY-NC-ND Creative Commons License. Except where otherwise noted, this item's license is described as © 2019, American Mathematical Society. First published in Transactions of the American Mathematical Society 372(2), July 2019. This author's draft version of the work, after peer review, is reproduced under the CC BY-NC-ND Creative Commons License.
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